A point P lies inside a circle of radius 13\text{ cm} at a distance of 5\text{ cm} from th… — JAMB UTME

A point $P$ lies inside a circle of radius $13\text{ cm}$ at a distance of $5\text{ cm}$ from the center $O$ . A chord $AB$ passes through point $P$ such that the ratio of segment $AP$ to segment $PB$ is $1:4$ . Calculate the total length of the chord $AB$ in centimeters.

Answer: 30 cm

Answer

The total length of chord

AB

30\text{ cm}

Using the Power of a Point property for an interior point

P

, the product of the chord segments is

AP \cdot PB = R^2 - OP^2 = 13^2 - 5^2 = 144

. Given

AP : PB = 1 : 4

, we write

AP = x

and

PB = 4x

, leading to

4x^2 = 144 \implies x = 6\text{ cm}

. Summing the two segments gives

AB = 6 + 24 = 30\text{ cm}

Step-by-Step Solution

Calculate the constant product of chord segments passing through interior point

P

AP \cdot PB = R^2 - OP^2 = 13^2 - 5^2 = 169 - 25 = 144

By the intersecting chords theorem, the product of segments created by an interior point

P

on any chord equals

(R - d)(R + d) = R^2 - d^2

Set up an algebraic equation using the segment ratio

AP : PB = 1 : 4

Let

AP = x

and

PB = 4x

, giving

(x)(4x) = 144 \implies 4x^2 = 144

Expressing both chord segments in terms of a single variable

x

allows direct calculation of the segment lengths.

Solve for

x

to find the individual segment lengths.

x^2 = 36 \implies x = 6\text{ cm}

. Therefore,

AP = 6\text{ cm}

and

PB = 24\text{ cm}

Taking the positive square root gives the scale factor

x

since physical distances must be positive.

Sum the segment lengths to find the total chord length.

AB = AP + PB = 6 + 24 = 30\text{ cm}

The entire chord length is the sum of its two divided parts.

Key Concept

Intersecting Chords Theorem and Power of an Interior Point

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