NCERT Solutions for Class 11 Maths Chapter 9 Straight Lines

Let ABCD be the given quadrilateral with vertices A (-4,5), B (0,7), C (5.-5) and D (-4,-2).

Now, let us plot the points on the Cartesian plane by joining the points AB, BC, CD, and AD, which give us the required quadrilateral.

To find the area, draw diagonal AC.

So, area (ABCD) = area (∆ABC) + area (∆ADC)

Then, area of triangle with vertices (x1,y1) , (x2, y2) and (x3,y3) is

Are of ∆ ABC = ½ [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]

= ½ [-4 (7 + 5) + 0 (-5 – 5) + 5 (5 – 7)] unit2

= ½ [-4 (12) + 5 (-2)] unit2

= ½ (58) unit2

= 29 unit2

Are of ∆ ACD = ½ [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]

= ½ [-4 (-5 + 2) + 5 (-2 – 5) + (-4) (5 – (-5))] unit2

= ½ [-4 (-3) + 5 (-7) – 4 (10)] unit2

= ½ (-63) unit2

= -63/2 unit2

Since area cannot be negative, area ∆ ACD = 63/2 unit2

Area (ABCD) = 29 + 63/2

= 121/2 unit2

 

Let us consider ABC, the given equilateral triangle with side 2a.

Where, AB = BC = AC = 2a

In the above figure, assuming that the base BC lies on the x-axis such that the mid-point of BC is at the origin, i.e., BO = OC = a, where O is the origin.

The coordinates of point C are (0, a) and that of B are (0,-a).

The line joining a vertex of an equilateral ∆ with the mid-point of its opposite side is perpendicular.

So, vertex A lies on the y –axis.

By applying Pythagoras’ theorem,

(AC)2 = OA2 + OC2

(2a)2= a2 + OC2

4a2 – a2 = OC2

3a2 = OC2

OC =√3a

Co-ordinates of point C = ± √3a, 0

∴ The vertices of the given equilateral triangle are (0, a), (0, -a), (√3a, 0)

Or (0, a), (0, -a) and (-√3a, 0)

 

Given:

Points P (x1, y1) and Q(x2, y2)

(i) When PQ is parallel to the y-axis, then x1 = x2

So, the distance between P and Q is given by

= |y2 – y1|

(ii) When PQ is parallel to the x-axis, then y1 = y2

So, the distance between P and Q is given by =

=

= |x2 – x1|

 

Let us consider (a, 0) to be the point on the x-axis that is equidistant from the point (7, 6) and (3, 4).

So,

Now, let us square on both sides; we get,

a2 – 14a + 85 = a2 – 6a + 25

-8a = -60

a = 60/8

= 15/2

∴ The required point is (15/2, 0)

 

The co-ordinates of the mid-point of the line segment joining the points P (0, – 4) and B (8, 0) are (0+8)/2, (-4+0)/2 = (4, -2)

The slope ‘m’ of the line non-vertical line passing through the point (x1, y1) and

(x2, y2) is given by m = (y2 – y1)/(x2 – x1) where, x ≠ x1

The slope of the line passing through (0, 0) and (4, -2) is (-2-0)/(4-0) = -1/2

∴ The required slope is -1/2.

 

The vertices of the given triangle are (4, 4), (3, 5) and (–1, –1).

The slope (m) of the line non-vertical line passing through the point (x1, y1) and

(x2, y2) is given by m = (y2 – y1)/(x2 – x1) where, x ≠ x1

So, the slope of the line AB (m1) = (5-4)/(3-4) = 1/-1 = -1

The slope of the line BC (m2) = (-1-5)/(-1-3) = -6/-4 = 3/2

The slope of the line CA (m3) = (4+1)/(4+1) = 5/5 = 1

It is observed that m1.m3 = -1.1 = -1

Hence, the lines AB and CA are perpendicular to each other.

∴ given triangle is right-angled at A (4, 4)

And the vertices of the right-angled ∆ are (4, 4), (3, 5) and (-1, -1)

 

We know that if a line makes an angle of 30° with the positive direction of the y-axis measured anti-clock-wise, then the angle made by the line with the positive direction of the x-axis measured anti-clock-wise is 90° + 30° = 120°

∴ The slope of the given line is tan 120° = tan (180° – 60°)

= – tan 60°

= –√3

 

If the points (x, – 1), (2, 1) and (4, 5) are collinear, then the Slope of AB = Slope of BC

Then, (1+1)/(2-x) = (5-1)/(4-2)

2/(2-x) = 4/2

2/(2-x) = 2

2 = 2(2-x)

2 = 4 – 2x

2x = 4 – 2

2x = 2

x = 2/2

= 1

∴ The required value of x is 1.

 

 

Let the given point be A (-2, -1) , B (4, 0) , C ( 3, 3) and D ( -3, 2)

So now, the slope of AB = (0+1)/(4+2) = 1/6

The slope of CD = (3-2)/(3+3) = 1/6

Hence, the Slope of AB = Slope of CD

∴ AB ∥ CD

Now,

The slope of BC = (3-0)/(3-4) = 3/-1 = -3

The slope of AD = (2+1)/(-3+2) = 3/-1 = -3

Hence, the Slope of BC = Slope of AD

∴ BC ∥ AD

Thus, the pair of opposite sides are quadrilateral are parallel, so we can say that ABCD is a parallelogram.

Hence, the given vertices, A (-2, -1), B (4, 0), C(3, 3) and D(-3, 2) are vertices of a parallelogram.

 

The Slope of the line joining the points (3, -1) and (4, -2) is given by

m = (y2 – y1)/(x2 – x1) where, x ≠ x1

m = (-2 –(-1))/(4-3)

= (-2+1)/(4-3)

= -1/1

= -1

The angle of inclination of the line joining the points (3, -1) and (4, -2) is given by

tan θ = -1

θ = (90° + 45°) = 135°

∴ The angle between the x-axis and the line joining the points (3, –1) and (4, –2) is 135°.

 

Let us consider ‘m1’ and ‘m’ be the slope of the two given lines such that m1 = 2m

We know that if θ is the angle between the lines l1 and l2 with slope m1 and m2, then

1+2m2 = -3m

2m2 +1 +3m = 0

2m (m+1) + 1(m+1) = 0

(2m+1) (m+1)= 0

m = -1 or -1/2

If m = -1, then the slope of the lines are -1 and -2

If m = -1/2, then the slope of the lines are -1/2 and -1

Case 2:

2m2 – 3m + 1 = 0

2m2 – 2m – m + 1 = 0

2m (m – 1) – 1(m – 1) = 0

m = 1 or 1/2

If m = 1, then the slope of the lines are 1 and 2

If m = 1/2, then the slope of the lines are 1/2 and 1

∴ The slope of the lines are [-1 and -2] or [-1/2 and -1] or [1 and 2] or [1/2 and 1]

 

Given: the slope of the line is ‘m’.

The slope of the line passing through (x1, y1) and (h, k) is (k – y1)/(h – x1)

So,

(k – y1)/(h – x1)  = m

(k – y1) = m (h – x1)

Hence, proved.

 

Let us consider if the given points A (h, 0), B (a, b) and C (0, k) lie on a line.

Then, the slope of AB = slope of BC

(b – 0)/(a – h) = (k – b)/(0 – a)

By simplifying, we get

-ab = (k-b) (a-h)

-ab = ka- kh –ab +bh

ka +bh = kh

Divide both sides by kh; we get

ka/kh + bh/kh = kh/kh

a/h + b/k = 1

Hence, proved.

 

We know that line AB passes through points A (1985, 92) and B (1995, 97).

Its slope will be (97 – 92)/(1995 – 1985) = 5/10 = 1/2

Let ‘y’ be the population in the year 2010. Then, according to the given graph, AB must pass through point C (2010, y)

So now, slope of AB = slope of BC

15/2 = y – 97

y = 7.5 + 97 = 104.5

∴ The slope of line AB is 1/2, while in the year 2010, the population will be 104.5 crores

 

The y-coordinate of every point on the x-axis is 0.

∴ The equation of the x-axis is y = 0.

The x-coordinate of every point on the y-axis is 0.

∴ The equation of the y-axis is y = 0.

 

Given:

Point (-4, 3) and slope, m = 1/2

We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0) only if its coordinates satisfy the equation y – y0 = m (x – x0)

So, y – 3 = 1/2 (x – (-4))

y – 3 = 1/2 (x + 4)

2(y – 3) = x + 4

2y – 6 = x + 4

x + 4 – (2y – 6) = 0

x + 4 – 2y + 6 = 0

x – 2y + 10 = 0

∴ The equation of the line is x – 2y + 10 = 0

 

Given:

Point (0, 0) and slope, m = m

We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0) only if its coordinates satisfy the equation y – y0 = m (x – x0)

So, y – 0 = m (x – 0)

y = mx

y – mx = 0

∴ The equation of the line is y – mx = 0

 

Given: point (2, 2√3) and θ = 75°

Equation of line: (y – y1) = m (x – x1)

where, m = slope of line = tan θ and (x1, y1) are the points through which line passes

∴ m = tan 75°

75° = 45° + 30°

Applying the formula:

We know that the point (x, y) lies on the line with slope m through the fixed point (x1, y1), only if its coordinates satisfy the equation y – y1 = m (x – x1)

Then, y – 2√3 = (2 + √3) (x – 2)

y – 2√3 = 2 x – 4 + √3 x – 2 √3

y = 2 x – 4 + √3 x

(2 + √3) x – y – 4 = 0

∴ The equation of the line is (2 + √3) x – y – 4 = 0

 

Given:

Slope, m = -2

We know that if a line L with slope m makes x-intercept d, then the equation of L is

y = m(x − d).

If the distance is 3 units to the left of the origin, then d = -3

So, y = (-2) (x – (-3))

y = (-2) (x + 3)

y = -2x – 6

2x + y + 6 = 0

∴ The equation of the line is 2x + y + 6 = 0

 

Given: θ = 30°

We know that slope, m = tan θ

m = tan30° = (1/√3)

We know that the point (x, y) on the line with slope m and y-intercept c lies on the line only if y = mx + c

If the distance is 2 units above the origin, c = +2

So, y = (1/√3)x + 2

y = (x + 2√3) / √3

√3 y = x + 2√3

x – √3 y + 2√3 = 0

∴ The equation of the line is x – √3 y + 2√3 = 0

 

Given:

Points (-1, 1) and (2, -4)

We know that the equation of the line passing through the points (x1, y1) and (x2, y2) is given by

y – 1 = -5/3 (x + 1)

3 (y – 1) = (-5) (x + 1)

3y – 3 = -5x – 5

3y – 3 + 5x + 5 = 0

5x + 3y + 2 = 0

∴ The equation of the line is 5x + 3y + 2 = 0

 

Given: p = 5 and ω = 30°

We know that the equation of the line having normal distance p from the origin and angle ω, which the normal makes with the positive direction of the x-axis, is given by x cos ω + y sin ω = p.

Substituting the values in the equation, we get

x cos30° + y sin30° = 5

x(√3 / 2) + y( 1/2 ) = 5

√3 x + y = 5(2) = 10

√3 x + y – 10 = 0

∴ The equation of the line is √3 x + y – 10 = 0

 

Vertices of ΔPQR, i.e., P (2, 1), Q (-2, 3) and R (4, 5)

Let RL be the median of vertex R.

So, L is a midpoint of PQ.

We know that the midpoint formula is given by
.

∴ L = = (0, 2)

We know that the equation of the line passing through the points (x1, y1) and (x2, y2) is given by

y – 5 = -3/-4 (x-4)

(-4) (y – 5) = (-3) (x – 4)

-4y + 20 = -3x + 12

-4y + 20 + 3x – 12 = 0

3x – 4y + 8 = 0

∴ The equation of median through the vertex R is 3x – 4y + 8 = 0

 

Given:

Points are (2, 5) and (-3, 6).

We know that slope, m = (y2 – y1)/(x2 – x1)

= (6 – 5)/(-3 – 2)

= 1/-5 = -1/5

We know that two non-vertical lines are perpendicular to each other only if their slopes are negative reciprocals of each other.

Then, m = (-1/m)

= -1/(-1/5)

= 5

We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0), only if its coordinates satisfy the equation y – y0 = m (x – x0)

Then, y – 5 = 5(x – (-3))

y – 5 = 5x + 15

5x + 15 – y + 5 = 0

5x – y + 20 = 0

∴ The equation of the line is 5x – y + 20 = 0

 

We know that the coordinates of a point dividing the line segment joining the points (x1, y1) and (x2, y2) internally in the ratio m: n are

We know that slope, m = (y2 – y1)/(x2 – x1)

= (3 – 0)/(2 – 1)

= 3/1

= 3

We know that two non-vertical lines are perpendicular to each other only if their slopes are negative reciprocals of each other.

Then, m = (-1/m) = -1/3

We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0), only if its coordinates satisfy the equation y – y0 = m (x – x0)

Here, the point is

3((1 + n) y – 3) = (-(1 + n) x + 2 + n)

3(1 + n) y – 9 = – (1 + n) x + 2 + n

(1 + n) x + 3(1 + n) y – n – 9 – 2 = 0

(1 + n) x + 3(1 + n) y – n – 11 = 0

∴ The equation of the line is (1 + n) x + 3(1 + n) y – n – 11 = 0

 

Given: the line cuts off equal intercepts on the coordinate axes, i.e., a = b

We know that equation of the line intercepts a and b on the x-and the y-axis, respectively, which is

x/a + y/b = 1

So, x/a + y/a = 1

x + y = a … (1)

Given: point (2, 3)

2 + 3 = a

a = 5

Substitute value of ‘a’ in (1), we get

x + y = 5

x + y – 5 = 0

∴ The equation of the line is x + y – 5 = 0

 

We know that equation of the line-making intercepts a and b on the x-and the y-axis, respectively, is x/a + y/b = 1 . … (1)

Given: sum of intercepts = 9

a + b = 9

b = 9 – a

Now, substitute the value of b in the above equation, and we get

x/a + y/(9 – a) = 1

Given: the line passes through point (2, 2)

So, 2/a + 2/(9 – a) = 1

[2(9 – a) + 2a] / a(9 – a) = 1

[18 – 2a + 2a] / a(9 – a) = 1

18/a(9 – a) = 1

18 = a (9 – a)

18 = 9a – a2

a2 – 9a + 18 = 0

Upon factorising, we get

a2 – 3a – 6a + 18 = 0

a (a – 3) – 6 (a – 3) = 0

(a – 3) (a – 6) = 0

a = 3 or a = 6

Let us substitute in (1)

Case 1 (a = 3):

Then b = 9 – 3 = 6

x/3 + y/6 = 1

2x + y = 6

2x + y – 6 = 0

Case 2 (a = 6):

Then b = 9 – 6 = 3

x/6 + y/3 = 1

x + 2y = 6

x + 2y – 6 = 0

∴ The equation of the line is 2x + y – 6 = 0 or x + 2y – 6 = 0

 

Given:

Point (0, 2) and θ = 2π/3

We know that m = tan θ

m = tan (2π/3) = -√3

We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0), only if its coordinates satisfy the equation y – y0 = m (x – x0)

y – 2 = -√3 (x – 0)

y – 2 = -√3 x

√3 x + y – 2 = 0

Given, the equation of the line parallel to the above-obtained equation crosses the y-axis at a distance of 2 units below the origin.

So, the point = (0, -2) and m = -√3

From point slope form equation,

y – (-2) = -√3 (x – 0)

y + 2 = -√3 x

√3 x + y + 2 = 0

∴ The equation of the line is √3 x + y – 2 = 0, and the line parallel to it is √3 x + y + 2 = 0

 

Given:

Points are origin (0, 0) and (-2, 9).

We know that slope, m = (y2 – y1)/(x2 – x1)

= (9 – 0)/(-2-0)

= -9/2

We know that two non-vertical lines are perpendicular to each other only if their slopes are negative reciprocals of each other.

m = (-1/m) = -1/(-9/2) = 2/9

We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0) only if its coordinates satisfy the equation y – y0 = m (x – x0)

y – 9 = (2/9) (x – (-2))

9(y – 9) = 2(x + 2)

9y – 81 = 2x + 4

2x + 4 – 9y + 81 = 0

2x – 9y + 85 = 0

∴ The equation of the line is 2x – 9y + 85 = 0

 

Let us assume ‘L’ along X-axis and ‘C’ along Y-axis; we have two points (124.942, 20) and (125.134, 110) in XY-plane.

We know that the equation of the line passing through the points (x1, y1) and (x2, y2) is given by

 

Assuming the relationship between the selling price and demand is linear.

Let us assume the selling price per litre along X-axis and demand along Y-axis, we have two points (14, 980) and (16, 1220) in XY-plane.

We know that the equation of the line passing through the points (x1, y1) and (x2, y2) is given by

y – 980 = 120 (x – 14)

y = 120 (x – 14) + 980

When x = Rs 17/litre,

y = 120 (17 – 14) + 980

y = 120(3) + 980

y = 360 + 980 = 1340

∴ The owner can sell 1340 litres weekly at Rs. 17/litre.

 

Let AB be a line segment whose midpoint is P (a, b).

Let the coordinates of A and B be (0, y) and (x, 0), respectively.

a (y – 2b) = -bx

ay – 2ab = -bx

bx + ay = 2ab

Divide both sides with ab, then

Hence, proved.

 

Let us consider AB to be the line segment, such that r (h, k) divides it in the ratio 1: 2.

So, the coordinates of A and B be (0, y) and (x, 0), respectively.

We know that the coordinates of a point dividing the line segment join the points (x1, y1) and (x2, y2) internally in the ratio m: n is

h = 2x/3 and k = y/3

x = 3h/2 and y = 3k

∴ A = (0, 3k) and B = (3h/2, 0)

We know that the equation of the line passing through the points (x1, y1) and (x2, y2) is given by

3h(y – 3k) = -6kx

3hy – 9hk = -6kx

6kx + 3hy = 9hk

Let us divide both sides by 9hk, and we get,

2x/3h + y/3k = 1

∴ The equation of the line is given by 2x/3h + y/3k = 1

 

According to the question,

If we have to prove that the given three points (3, 0), (– 2, – 2) and (8, 2) are collinear, then we have to also prove that the line passing through the points (3, 0) and (– 2, – 2) also passes through the point (8, 2).

By using the formula,

The equation of the line passing through the points (x1, y1) and (x2, y2) is given by

-5y = -2 (x – 3)

-5y = -2x + 6

2x – 5y = 6

If 2x – 5y = 6 passes through (8, 2),

2x – 5y = 2(8) – 5(2)

= 16 – 10

= 6

= RHS

The line passing through points (3, 0) and (– 2, – 2) also passes through the point (8, 2).

Hence, proved. The given three points are collinear.

 

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(i) x + 7y = 0

Given:

The equation is x + 7y = 0

The slope-intercept form is represented in the form ‘y = mx + c’, where m is the slope and c is the y-intercept.

So, the above equation can be expressed as

y = -1/7x + 0

∴ The above equation is of the form y = mx + c, where m = -1/7 and c = 0

(ii) 6x + 3y – 5 = 0

Given:

The equation is 6x + 3y – 5 = 0

The slope-intercept form is represented in the form ‘y = mx + c’, where m is the slope and c is the y-intercept.

So, the above equation can be expressed as

3y = -6x + 5

y = -6/3x + 5/3

= -2x + 5/3

∴ The above equation is of the form y = mx + c, where m = -2 and c = 5/3

(iii) y = 0

Given:

The equation is y = 0

The slope-intercept form is given by ‘y = mx + c’, where m is the slope and c is the y-intercept.

y = 0 × x + 0

∴ The above equation is of the form y = mx + c, where m = 0 and c = 0

 

(i) 3x + 2y – 12 = 0

Given:

The equation is 3x + 2y – 12 = 0

The equation of the line in intercept form is given by x/a + y/b = 1, where ‘a’ and ‘b’ are intercepted on the x-axis and the y-axis, respectively.

So, 3x + 2y = 12

Now, let us divide both sides by 12; we get

3x/12 + 2y/12 = 12/12

x/4 + y/6 = 1

∴ The above equation is of the form x/a + y/b = 1, where a = 4, b = 6

The intercept on the x-axis is 4.

The intercept on the y-axis is 6.

(ii) 4x – 3y = 6

Given:

The equation is 4x – 3y = 6

The equation of the line in intercept form is given by x/a + y/b = 1, where ‘a’ and ‘b’ are intercepted on the x-axis and the y-axis, respectively.

So, 4x – 3y = 6

Now, let us divide both sides by 6; we get

4x/6 – 3y/6 = 6/6

2x/3 – y/2 = 1

x/(3/2) + y/(-2) = 1

∴ The above equation is of the form x/a + y/b = 1, where a = 3/2, b = -2

The intercept on the x-axis is 3/2.

The intercept on the y-axis is -2.

(iii) 3y + 2 = 0

Given:

The equation is 3y + 2 = 0

The equation of the line in intercept form is given by x/a + y/b = 1, where ‘a’ and ‘b’ are intercepted on the x-axis and the y-axis, respectively.

So, 3y = -2

Now, let us divide both sides by -2; we get

3y/-2 = -2/-2

3y/-2 = 1

y/(-2/3) = 1

∴ The above equation is of the form x/a + y/b = 1, where a = 0, b = -2/3

The intercept on the x-axis is 0.

The intercept on the y-axis is -2/3.

 

(i) x – √3y + 8 = 0

Given:

The equation is x – √3y + 8 = 0

The equation of the line in normal form is given by x cos θ + y sin θ = p where ‘θ’ is the angle between the perpendicular and the positive x-axis and ‘p’ is the perpendicular distance from the origin.

So now, x – √3y + 8 = 0

x – √3y = -8

Divide both the sides by √(12 + (√3)2) = √(1 + 3) = √4 = 2

x/2 – √3y/2 = -8/2

(-1/2)x + √3/2y = 4

This is in the form of: x cos 120o + y sin 120o = 4

∴ The above equation is of the form x cos θ + y sin θ = p, where θ = 120° and p = 4.

Perpendicular distance of the line from origin = 4

The angle between the perpendicular and positive x-axis = 120°

(ii) y – 2 = 0

Given:

The equation is y – 2 = 0

The equation of the line in normal form is given by x cos θ + y sin θ = p where ‘θ’ is the angle between the perpendicular and the positive x-axis and ‘p’ is the perpendicular distance from the origin.

So now, 0 × x + 1 × y = 2

Divide both sides by √(02 + 12) = √1 = 1

0 (x) + 1 (y) = 2

This is in the form of: x cos 90o + y sin 90o = 2

∴ The above equation is of the form x cos θ + y sin θ = p, where θ = 90° and p = 2.

Perpendicular distance of the line from origin = 2

The angle between the perpendicular and positive x-axis = 90°

(iii) x – y = 4

Given:

The equation is x – y + 4 = 0

The equation of the line in normal form is given by x cos θ + y sin θ = p where ‘θ’ is the angle between the perpendicular and the positive x-axis and ‘p’ is the perpendicular distance from the origin.

So now, x – y = 4

Divide both the sides by √(12 + 12) = √(1+1) = √2

x/√2 – y/√2 = 4/√2

(1/√2)x + (-1/√2)y = 2√2

This is in the form: x cos 315o + y sin 315o = 2√2

∴ The above equation is of the form x cos θ + y sin θ = p, where θ = 315° and p = 2√2.

Perpendicular distance of the line from origin = 2√2

The angle between the perpendicular and the positive x-axis = 315°

 

Given:

The equation of the line is 12(x + 6) = 5(y – 2).

12x + 72 = 5y – 10

12x – 5y + 82 = 0 … (1)

Now, compare equation (1) with the general equation of line Ax + By + C = 0, where A = 12, B = –5, and C = 82

Perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by

∴ The distance is 5 units.

 

Given:

The equation of the line is x/3 + y/4 = 1

4x + 3y = 12

4x + 3y – 12 = 0 …. (1)

Now, compare equation (1) with the general equation of line Ax + By + C = 0, where A = 4, B = 3, and C = -12

Let (a, 0) be the point on the x-axis whose distance from the given line is 4 units.

So, the perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by

|4a – 12| = 4 × 5

± (4a – 12) = 20

4a – 12 = 20 or – (4a – 12) = 20

4a = 20 + 12 or 4a = -20 + 12

a = 32/4 or a = -8/4

a = 8 or a = -2

∴ The required points on the x-axis are (-2, 0) and (8, 0)

 

(i) 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0

Given:

The parallel lines are 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0.

By using the formula,

The distance (d) between parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0 is given by

∴ The distance between parallel lines is 65/17

(ii) l(x + y) + p = 0 and l (x + y) – r = 0

Given:

The parallel lines are l (x + y) + p = 0 and l (x + y) – r = 0

lx + ly + p = 0 and lx + ly – r = 0

By using the formula,

The distance (d) between parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0 is given by

∴ The distance between parallel lines is |p+r|/l√2

 

Given:

The line is 3x – 4y + 2 = 0

So, y = 3x/4 + 2/4

= 3x/4 + ½

Which is of the form y = mx + c, where m is the slope of the given line.

The slope of the given line is 3/4

We know that parallel lines have the same slope.

∴ Slope of other line = m = 3/4

The equation of line having slope m and passing through (x1, y1) is given by

y – y1 = m (x – x1)

∴ The equation of the line having slope 3/4 and passing through (-2, 3) is

y – 3 = ¾ (x – (-2))

4y – 3 × 4 = 3x + 3 × 2

3x – 4y = 18

∴ The equation is 3x – 4y = 18

 

Given:

The equation of line is x – 7y + 5 = 0

So, y = 1/7x + 5/7 [which is of the form y = mx + c, where m is the slope of the given line.]

The slope of the given line is 1/7

The slope of the line perpendicular to the line having slope m is -1/m

The slope of the line perpendicular to the line having a slope of 1/7 is -1/(1/7) = -7

So, the equation of the line with slope -7 and the x-intercept 3 is given by y = m(x – d)

y = -7 (x – 3)

y = -7x + 21

7x + y = 21

∴ The equation is 7x + y = 21

 

Given:

The lines are √3x + y = 1 and x + √3y = 1

So, y = -√3x + 1 … (1) and

y = -1/√3x + 1/√3 …. (2)

The slope of the line (1) is m1 = -√3, while the slope of the line (2) is m2 = -1/√3

Let θ be the angle between two lines.

So,

θ = 30°

∴ The angle between the given lines is either 30° or 180°- 30° = 150°

 

Let the slope of the line passing through (h, 3) and (4, 1) be m1

Then, m1 = (1-3)/(4-h) = -2/(4-h)

Let the slope of line 7x – 9y – 19 = 0 be m2

7x – 9y – 19 = 0

So, y = 7/9x – 19/9

m2 = 7/9

Since the given lines are perpendicular,

m1 × m2 = -1

-2/(4-h) × 7/9 = -1

-14/(36-9h) = -1

-14 = -1 × (36 – 9h)

36 – 9h = 14

9h = 36 – 14

h = 22/9

∴ The value of h is 22/9

 

Let the slope of line Ax + By + C = 0 be m

Ax + By + C = 0

So, y = -A/Bx – C/B

m = -A/B

By using the formula,

Equation of the line passing through point (x1, y1) and having slope m = -A/B is

y – y1 = m (x – x1)

y – y1= -A/B (x – x1)

B (y – y1) = -A (x – x1)

∴ A(x – x1) + B(y – y1) = 0

So, the line through point (x1, y1) and parallel to the line Ax + By + C = 0 is A (x – x1) + B (y – y1) = 0

Hence, proved.

 

Given: m1 = 2

Let the slope of the first line be m1

And let the slope of the other line be m2.

The angle between the two lines is 60°.

So,

 

Given:

The right bisector of a line segment bisects the line segment at 90°.

End-points of the line segment AB are given as A (3, 4) and B (–1, 2).

Let the mid-point of AB be (x, y).

x = (3-1)/2= 2/2 = 1

y = (4+2)/2 = 6/2 = 3

(x, y) = (1, 3)

Let the slope of line AB be m1

m1 = (2 – 4)/(-1 – 3)

= -2/(-4)

= 1/2

And let the slope of the line perpendicular to AB be m2

m2 = -1/(1/2)

= -2

The equation of the line passing through (1, 3) and having a slope of –2 is

(y – 3) = -2 (x – 1)

y – 3 = – 2x + 2

2x + y = 5

∴ The required equation of the line is 2x + y = 5

 

Let us consider the coordinates of the foot of the perpendicular from (-1, 3) to the line 3x – 4y – 16 = 0 be (a, b)

So, let the slope of the line joining (-1, 3) and (a, b) be m1

m1 = (b-3)/(a+1)

And let the slope of the line 3x – 4y – 16 = 0 be m2

y = 3/4x – 4

m2 = 3/4

Since these two lines are perpendicular, m1 × m2 = -1

(b-3)/(a+1) × (3/4) = -1

(3b-9)/(4a+4) = -1

3b – 9 = -4a – 4

4a + 3b = 5 …….(1)

Point (a, b) lies on the line 3x – 4y = 16

3a – 4b = 16 ……..(2)

Solving equations (1) and (2), we get

a = 68/25 and b = -49/25

∴ The coordinates of the foot of perpendicular are (68/25, -49/25)

 

Given:

The perpendicular from the origin meets the given line at (–1, 2).

The equation of the line is y = mx + c

The line joining the points (0, 0) and (–1, 2) is perpendicular to the given line.

So, the slope of the line joining (0, 0) and (–1, 2) = 2/(-1) = -2

The slope of the given line is m.

m × (-2) = -1

m = 1/2

Since point (-1, 2) lies on the given line,

y = mx + c

2 = 1/2 × (-1) + c

c = 2 + 1/2 = 5/2

∴ The values of m and c are 1/2 and 5/2, respectively.

 

Given:

The equations of the given lines are

x cos θ – y sin θ = k cos 2θ …………………… (1)

x sec θ + y cosec θ = k ……………….… (2)

Perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by

q = k cos θ sin θ

Multiply both sides by 2, and we get

2q = 2k cos θ sin θ = k × 2sin θ cos θ

2q = k sin 2θ

Squaring both sides, we get

4q2 = k2 sin22θ …………………(4)

Now add (3) and (4); we get

p2 + 4q2 = k2 cos2 2θ + k2 sin2 2θ

p2 + 4q2 = k2 (cos2 2θ + sin2 2θ) [Since, cos2 2θ + sin2 2θ = 1]

∴ p2 + 4q2 = k2

Hence proved.

 

Let AD be the altitude of triangle ABC from vertex A.

So, AD is perpendicular to BC.

Given:

Vertices A (2, 3), B (4, –1) and C (1, 2)

Let the slope of the line BC = m1

m1 = (- 1 – 2)/(4 – 1)

m1 = -1

Let the slope of the line AD be m2

AD is perpendicular to BC.

m1 × m2 = -1

-1 × m2 = -1

m2 = 1

The equation of the line passing through the point (2, 3) and having a slope of 1 is

y – 3 = 1 × (x – 2)

y – 3 = x – 2

y – x = 1

Equation of the altitude from vertex A = y – x = 1

Length of AD = Length of the perpendicular from A (2, 3) to BC

The equation of BC is

y + 1 = -1 × (x – 4)

y + 1 = -x + 4

x + y – 3 = 0 …………………(1)

Perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by

Now, compare equation (1) to the general equation of the line, i.e., Ax + By + C = 0; we get

Length of AD =

[where, A = 1, B = 1 and C = -3]

∴ The equation and the length of the altitude from vertex A are y – x = 1 and

√2 units, respectively.

 

The equation of a line whose intercepts on the axes are a and b is x/a + y/b = 1

bx + ay = ab

bx + ay – ab = 0 ………………..(1)

Perpendicular distance (d) of a line Ax + By + C = 0 from a point (x1, y1) is given by

Now, square on both sides; we get

∴ 1/p2 = 1/a2 + 1/b2

Hence, proved.

 

It is given that

(k – 3) x – (4 – k2) y + k2 – 7k + 6 = 0 … (1)

(a) Here, if the line is parallel to the x-axis

Slope of the line = Slope of the x-axis

It can be written as

(4 – k2) y = (k – 3) x + k2 – 7k + 6 = 0

We get

By further calculation,

k – 3 = 0

k = 3

Hence, if the given line is parallel to the x-axis, then the value of k is 3.

(b) Here, if the line is parallel to the y-axis, it is vertical, and the slope will be undefined.

So, the slope of the given line

k2 = 4

k = ± 2

Hence, if the given line is parallel to the y-axis, then the value of k is ± 2.

(c) Here, if the line is passing through (0, 0), which is the origin satisfies the given equation of the line.

(k – 3) (0) – (4 – k2) (0) + k2 – 7k + 6 = 0

By further calculation,

k2 – 7k + 6 = 0

Separating the terms,

k2 – 6k – k + 6 = 0

We get

(k – 6) (k – 1) = 0

k = 1 or 6

Hence, if the given line is passing through the origin, then the value of k is either 1 or 6.

 

 

Consider the intercepts cut by the given lines on the a and b axes.

a + b = 1 …… (1)

ab = – 6 …….. (2)

By solving both equations, we get

a = 3 and b = -2 or a = – 2 and b = 3

We know that the equation of the line whose intercepts on the a and b axes is

Case I – a = 3 and b = – 2

So, the equation of the line is – 2x + 3y + 6 = 0, i.e. 2x – 3y = 6

Case II – a = -2 and b = 3

So, the equation of the line is 3x – 2y + 6 = 0, i.e. -3x + 2y = 6

Hence, the required equation of the lines are 2x – 3y = 6 and -3x + 2y = 6

 

Consider (0, b) as the point on the y-axis whose distance from line x/3 + y/4 = 1 is 4 units.

It can be written as 4x + 3y – 12 = 0 ……. (1)

By comparing equation (1) to the general equation of line Ax + By + C = 0, we get

A = 4, B = 3 and C = – 12

We know that the perpendicular distance (d) of a line Ax + By + C = 0 from (x1, y1) is written as

By cross multiplication,

20 = |3b – 12|

We get

20 = ± (3b – 12)

Here, 20 = (3b – 12) or 20 = – (3b – 12)

It can be written as

3b = 20 + 12 or 3b = -20 + 12

So, we get

b = 32/3 or b = -8/3

Hence, the required points are (0, 32/3) and (0, -8/3).

 

 

Here, the equation of any line parallel to the y-axis is of the form

x = a ……. (1)

Two given lines are

x – 7y + 5 = 0 …… (2)

3x + y = 0 …… (3)

By solving equations (2) and (3), we get

x = -5/22 and y = 15/22

(-5/ 22, 15/22) is the point of intersection of lines (2) and (3)

If the line x = a passes through point (-5/22, 15/22), we get a = -5/22

Hence, the required equation of the line is x = -5/22

 

It is given that

x/4 + y/6 = 1

We can write it as

3x + 2y – 12 = 0

So, we get

y = -3/2 x + 6, which is of the form y = mx + c

Here, the slope of the given line = -3/2

So, the slope of line perpendicular to the given line = -1/ (-3/2) = 2/3

Consider the given line intersects, the y-axis at (0, y)

By substituting x as zero in the equation of the given line,

y/6 = 1

y = 6

Hence, the given line intersects the y-axis at (0, 6).

We know that the equation of the line that has a slope of 2/3 and passes through the point (0, 6) is

(y – 6) = 2/3 (x – 0)

By further calculation,

3y – 18 = 2x

So, we get

2x – 3y + 18 = 0

Hence, the required equation of the line is 2x – 3y + 18 = 0

 

It is given that

y – x = 0 …… (1)

x + y = 0 …… (2)

x – k = 0 ……. (3)

Here, the point of intersection of

Lines (1) and (2) is

x = 0 and y = 0

Lines (2) and (3) is

x = k and y = – k

Lines (3) and (1) is

x = k and y = k

So, the vertices of the triangle formed by the three given lines are (0, 0), (k, -k) and (k, k).

Here, the area of triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is

½ |x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)|

So, the area of the triangle formed by the three given lines is

= ½ |0 (-k – k) + k (k – 0) + k (0 + k)| square units

By further calculation,

= ½ |k2 + k2| square units

So, we get

= ½ |2k2|

= k2 square units

 

It is given that

3x + y – 2 = 0 …… (1)

px + 2y – 3 = 0 ….. (2)

2x – y – 3 = 0 …… (3)

By solving equations (1) and (3), we get

x = 1 and y = -1

Here, the three lines intersect at one point, and the point of intersection of lines (1) and (3) will also satisfy line (2)

p (1) + 2 (-1) – 3 = 0

By further calculation,

p – 2 – 3 = 0

So we get

p = 5

Hence, the required value of p is 5.

 

It is given that

y = m1x + c1 ….. (1)

y = m2x + c2 ….. (2)

y = m3x + c3 ….. (3)

By subtracting equation (1) from (2), we get

0 = (m2 – m1) x + (c2 – c1)

(m1 – m2) x = c2 – c1

So we get

Taking out the common terms,

m1 (c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0

Therefore, m1 (c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0

 

Consider m1 as the slope of the required line

It can be written as

y = 1/2 x – 3/2 which is of the form y = mx + c

So, the slope of the given line m2 = 1/2

We know that the angle between the required line and line x – 2y = 3 is 45o

If θ is the acute angle between lines l1 and l2 with slopes m1 and m2,

It can be written as

2 + m1 = 1 – 2m1 or 2 + m1 = – 1 + 2m1

m1 = – 1/3 or m1 = 3

Case I – m1 = 3

Here, the equation of the line passing through (3, 2) and having a slope 3 is

y – 2 = 3 (x – 3)

By further calculation,

y – 2 = 3x – 9

So, we get

3x – y = 7

Case II – m1 = -1/3

Here, the equation of the line passing through (3, 2) and having a slope -1/3 is

y – 2 = – 1/3 (x – 3)

By further calculation,

3y – 6 = – x + 3

So, we get

x + 3y = 9

Hence, the equations of the lines are 3x – y = 7 and x + 3y = 9

 

Consider the equation of the line having equal intercepts on the axes as

x/a + y/a = 1

It can be written as

x + y = a ….. (1)

By solving equations 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0, we get

x = 1/13 and y = 5/13

(1/13, 5/13) is the point of intersection of two given lines.

We know that equation (1) passes through the point (1/13, 5/13).

1/13 + 5/13 = a

a = 6/13

So, equation (1) passes through (1/13, 5/13).

1/13 + 5/13 = a

We get

a = 6/13

Her, equation (1) becomes

x + y = 6/13

13x + 13y = 6

Hence, the required equation of the line is 13x + 13y = 6

 

Consider y = m1x as the equation of the line passing through the origin

 

 

By cross multiplication,

– k + 5 = 1 + k

We get

2k = 4

k = 2

Hence, the line joining the points (-1, 1) and (5, 7) is divided by the line x + y = 4 in the ratio 1: 2.

 

It is given that

2x – y = 0 ….. (1)

4x + 7y + 5 = 0 …… (2)

Here, A (1, 2) is a point on the line (1).

Consider B as the point of intersection of lines (1) and (2).

By solving equations (1) and (2), we get x = -5/18 and y = – 5/9

So, the coordinates of point B are (-5/18, -5/9).

From the distance formula, the distance between A and B

Hence, the required distance is
.

 

Consider y = mx + c as the line passing through the point (-1, 2).

So, we get

2 = m (-1) + c

By further calculation,

2 = -m + c

c = m + 2

Substituting the value of c

y = mx + m + 2 …… (1)

So the given line is

x + y = 4 ……. (2)

By solving both equations, we get

By cross multiplication,

1 + m2 = m2 + 1 + 2m

So, we get

2m = 0

m = 0

Hence, the slope of the required line must be zero, i.e., the line must be parallel to the x-axis.

 

Consider ABC as the right angles triangle where ∠C = 90o

Here, infinity such lines are present.

m is the slope of AC

So, the slope of BC = -1/m

Equation of AC –

y – 3 = m (x – 1)

By cross multiplication,

x – 1 = 1/m (y – 3)

Equation of BC –

y – 1 = – 1/m (x + 4)

By cross multiplication,

x + 4 = – m (y – 1)

By considering the values of m, we get

If m = 0,

So, we get

y – 3 = 0, x + 4 = 0

If m = ∞,

So, we get

x – 1 = 0, y – 1 = 0 we get x = 1, y = 1

 

It is given that

x + 3y = 7 ….. (1)

Consider B (a, b) as the image of point A (3, 8).

So line (1) is the perpendicular bisector of AB.

On further simplification,

a + 3b = – 13 ….. (3)

By solving equations (2) and (3), we get

a = – 1 and b = – 4

Hence, the image of the given point with respect to the given line is (-1, -4).

 

It is given that

y = 3x + 1 …… (1)

2y = x + 3 …… (2)

y = mx + 4 …… (3)

Here, the slopes of

Line (1), m1 = 3

Line (2), m2 = ½

Line (3), m3 = m

We know that lines (1) and (2) are equally inclined to line (3), which means that the angle between lines (1) and (3) equals the angle between lines (2) and (3).

On further calculation,

– m2 + m + 6 = 1 + m – 6m2

So, we get

5m2 + 5 = 0

Dividing the equation by 5,

m2 + 1 = 0

m = √-1, which is not real.

Therefore, this case is not possible.

If

 

 

In the same way, we can find the equation of the line for any signs of (x + y – 5) and (3x – 2y + 7)

Hence, point P must move on a line.

 

 

Here,

9h + 6k – 7 = 3 (3h + 2k + 6) or 9h + 6k – 7 = – 3 (3h + 2k + 6)

9h + 6k – 7 = 3 (3h + 2k + 6) is not possible as

9h + 6k – 7 = 3 (3h + 2k + 6)

By further calculation,

– 7 = 18 (which is wrong)

We know that

9h + 6k – 7 = -3 (3h + 2k + 6)

By multiplication,

9h + 6k – 7 = -9h – 6k – 18

We get

18h + 12k + 11 = 0

Hence, the required equation of the line is 18x + 12y + 11 = 0

 

Consider the coordinates of point A as (a, 0).

Construct a line (AL) which is perpendicular to the x-axis.

Here, the angle of incidence is equal to the angle of reflection

∠BAL = ∠CAL = Φ

∠CAX = θ

It can be written as

∠OAB = 180° – (θ + 2Φ) = 180° – [θ + 2(90° – θ)]

On further calculation,

= 180° – θ – 180° + 2θ

= θ

So, we get

∠BAX = 180° – θ

By cross multiplication,

3a – 3 = 10 – 2a

We get

a = 13/5

Hence, the coordinates of point A are (13/5, 0).

 

It is given that

We can write it as

bx cos θ + ay sin θ – ab = 0 ….. (1)

 

It is given that

2x – 3y + 4 = 0 …… (1)

3x + 4y – 5 = 0 ……. (2)

6x – 7y + 8 = 0 …… (3)

Here, the person is standing at the junction of the paths represented by lines (1) and (2).

By solving equations (1) and (2), we get

x = – 1/17 and y = 22/17

Hence, the person is standing at point (-1/17, 22/17).

We know that the person can reach path (3) in the least time if they walk along the perpendicular line to (3) from point (-1/17, 22/17)

Here, the slope of line (3) = 6/7

We get the slope of the line perpendicular to the line (3) = -1/ (6/7) = – 7/6

So, the equation of the line passing through (-1/17, 22/17) and having a slope of -7/6 is written as

By further calculation,

6 (17y – 22) = – 7 (17x + 1)

By multiplication,

102y – 132 = – 119x – 7

We get

1119x + 102y = 125

Therefore, the path that the person should follow is 119x + 102y = 125