CDS Mathematics · Circles

Intersecting Chords and Power of a Point

Through a fixed point, every line that meets the circle cuts off two segments with the same product.

All Circles notes CDS Mathematics strategy

Why this matters

Five PYQs, two of them HARD. One product answers them all: for chords crossing inside, AP × PB = CP × PD; for a secant and a tangent from outside, PA × PB = PT². The product also equals r² − d² inside and d² − r² outside.

Concept 1 of 1: Chords, secants and the tangent from one point

Measure along any line through

P

to the two points where it meets the circle, and multiply. The answer depends only on how far

P

is from the centre, so every line through

P

gives the same product.

Definition

Chords $AB$ and $CD$ crossing at $P$ inside: $AP \cdot PB = CP \cdot PD = r^2 - OP^2$ .
Secants $PAB$ and $PCD$ from $P$ outside: $PA \cdot PB = PC \cdot PD$ .
Secant and tangent from $P$ : $PA \cdot PB = PT^2 = OP^2 - r^2$ .
$PB$ is the WHOLE secant from $P$ ; the chord inside the circle is $AB = PB - PA$ .

Tangent and secant

PT^2 = PA \cdot PB

Worked example

Chords

AB

and

CD

cross at

P

, with

AP = 4

PB = 6

and

CP = 3

cm. Find

PD

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q86Moderate

Example 1 · Circles · Intersecting Chords and Power of a Point

Let

PAB

be a secant to a circle intersecting the circle at

A

and

B

. Let

PT

be the tangent segment. If

PA = 9

cm and

PT = 12

cm, then what is

AB

equal to?

The whole secant, then subtract

PT^2 = PA \cdot PB

PB

runs from

P

to the FAR point. The chord asked for is

PB - PA

; stopping at

PB

picks the distractor.

Given the product, the statements are not needed

AP \cdot PB

is already given,

CP \cdot PD

equals it by the theorem. In a data-sufficiency item, that means NEITHER statement is required.

Drill 4 more on chords, secants and the tangent from one point

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Chords, secants and the tangent from one point
Tangent and secant
$PT^2 = PA \cdot PB$

Watch out for (2)

The whole secant, then subtract
→ Chords, secants and the tangent from one point
Given the product, the statements are not needed
→ Chords, secants and the tangent from one point

Test yourself on Circles

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter test Drill all 5 questions