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Question

In the figure given below, PAB is a secant and PR is a tangent to the circle. If PA : AB = 1 : 8, and PR = 15 cm, find the length of PB.


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Solution

Given: PAB is a secant and PR is a tangent to the circle.

Also, PA : AB = 1 : 8 and PR = 15 cm.
Let, PA = x. Then AB = 8x
PR = 15 cm

If a chord and a tangent intersect externally, then the product of the lengths of the segmants of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.
i.e. (PA)×(PB)=(PR)2

(PA) (PA + AB) = (PR)2 ...[Since, PB = PA + AP]

(x) (x + 8x) = (15)2
(x) (9x) = 225
9x2=225
x2=2259
x2=25
x=5

PA = x = 5 cm
And AB = 8x = 8 x 5 = 40 cm
So, PB = PA + AB = 5 + 40 = 45 cm


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