Tangent-Secant Relationship

Definition: The tangent-secant relationship (also called the power of a point theorem) states:

PT² = PA × PB

Where:

• P is an external point.
• PT is the tangent from P to the circle (T is the point of tangency).
• PAB is a secant from P passing through the circle, with A and B on the circle (A is nearer to P).

Key facts:

• PT is the geometric mean of PA and PB.
• The relationship holds regardless of where the secant passes through the circle.
• If two secants are drawn instead, the relationship becomes: PA × PB = PC × PD.

Geometric Mean Interpretation:

• Since PT² = PA × PB, the tangent length PT is the geometric mean of PA and PB.
• Geometric mean of two numbers a and b = √(ab).
• This means: PT = √(PA × PB).
• The tangent length lies between PA and PB (since geometric mean ≤ arithmetic mean ≤ max value).

Why is it called "Power of a Point"?

• The value PT² = PA × PB is the same for every line through P that intersects the circle.
• This constant value depends only on the position of P relative to the circle (specifically, d² − r²).
• It is called the "power" because it characterises the geometric relationship between the point and the circle — much like a physical quantity characterises a system.
• The power is positive for external points, zero for points on the circle, and negative for internal points.

Proof of PT² = PA × PB:

Given: Circle with centre O, external point P, tangent PT, secant PAB.

1. Join OT, OA, OB, OP.
2. In △PTA and △PTB, consider the angles:
3. ∠TPB is common to both triangles.
4. ∠PTA = ∠TBA (tangent-chord angle = angle in alternate segment).
5. Therefore, △PTA ~ △PTB (AA similarity).
6. From similarity: PT/PA = PB/PT.
7. Cross-multiplying: PT² = PA × PB.

Key theorem used: The alternate segment theorem — the angle between a tangent and a chord at the point of tangency equals the angle in the alternate segment.

Alternate Proof Using Coordinates:

1. Place the circle at origin with radius r: x² + y² = r².
2. Let external point P = (d, 0) where d > r.
3. Tangent length: PT = √(d² − r²).
4. Secant along x-axis hits circle at A(r, 0) and B(−r, 0).
5. PA = d − r, PB = d + r.
6. PA × PB = (d − r)(d + r) = d² − r².
7. PT² = d² − r².
8. Therefore PT² = PA × PB. ✓

This coordinate proof confirms the geometric proof and shows the relationship holds for any secant direction (since the power of a point is independent of the line direction).

Configurations involving tangent-secant relationships:

Type 1: One tangent + One secant from the same point

• PT² = PA × PB
• Most common type in Class 10 problems.

Type 2: Two secants from the same point

• PA × PB = PC × PD
• No tangent involved.

Type 3: Two tangents from the same point

• PT₁ = PT₂ (tangents from an external point are equal).
• This is a special case: PT₁² = PT₂² (trivially true).

Intersecting Chords Theorem (for comparison):

• If two chords intersect at a point P inside the circle, then PA × PB = PC × PD.
• This is a different configuration from the tangent-secant theorem (which requires an external point).
• The "power of a point" is negative for internal points and positive for external points.

Unified Power of a Point:

• For any point P and any line through P intersecting a circle at A and B: the product PA × PB is constant.
• This constant is called the power of the point P with respect to the circle.
• Power = d² − r² (where d = distance from P to centre, r = radius).
• External point: power > 0, equals PT².
• Point on circle: power = 0 (tangent has length 0).
• Internal point: power < 0.

Steps to solve tangent-secant problems:

1. Draw a clear diagram showing the circle, external point, tangent, and secant.
2. Label all known lengths: PT (tangent), PA (near intersection), PB (far intersection).
3. Note: PB = PA + AB (where AB is the chord).
4. Apply PT² = PA × PB.
5. Solve for the unknown length.

Common problem types:

• Find tangent length: Given PA and PB → PT = √(PA × PB).
• Find secant segment: Given PT and PA → PB = PT²/PA.
• Find chord length: Given PT, PA → PB = PT²/PA → chord AB = PB − PA.

Problem-Solving Strategies:

• When given a tangent length and asked about a secant: use PT² = PA × PB to find the unknown secant segment.
• When given two secants: use PA × PB = PC × PD.
• When the secant passes through the centre: AB is a diameter, so PA = d − r and PB = d + r (where d = OP).
• When finding the radius: set up PT² = (d − r)(d + r) = d² − r², which gives r² = d² − PT².

Common Mistakes:

• Confusing PA with AB: PA is from the external point to the near intersection, not the chord length. PB = PA + AB.
• Forgetting which is nearer: A is always the intersection point closer to the external point P.
• Using the formula for internal points: PT² = PA × PB applies only to external points. For internal points, the intersecting chords theorem applies: PA × PB = PC × PD (different formula structure).

Optics:

• Light rays tangent to a circular lens and secant rays passing through it follow this geometric relationship, aiding lens design.

Surveying:

• Calculating distances to circular structures (tanks, silos) when tangent and secant measurements are available.

Gear Design:

• The contact geometry between gears involves tangent and secant lines to circular profiles.

Satellite Geometry:

• Line of sight from a satellite to the earth's surface is a tangent to Earth's circular cross-section. Communication signals passing through the atmosphere trace secant paths.

Architecture:

• Designing arch bridges requires tangent-secant calculations to determine the stress distribution at the point where a straight support meets a curved arch.

Robotics:

• Robot arm path planning around circular obstacles uses tangent and secant computations to find optimal paths.

Acoustics:

• Sound waves tangent to a circular amphitheatre wall produce specific acoustic effects. The tangent-secant relationship helps model reflection patterns.