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. [3 Marks]

Answer 1

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. $\left(PA\right)\times \left(PB\right)=(PR{)}^2$ [1 Mark]

(PA) (PA + AB) = $(PR{)}^2$ ...[Since, PB = PA + AP]

(x) (x + 8x) = $(15{)}^2$
(x) (9x) = 225
$9x^2=225$
$x^2=\frac{225}{9}$
$x^2=25$
$x=5$ [1 Mark]

$\therefore$ PA = x = 5 cm
And AB = 8x = 8 x 5 = 40 cm
So, PB = PA + AB = 5 + 40 = 45 cm [1 Mark]