Question

A and B are centres of circles of radii 9 cm and 2 cm such that AB = 17 cm and C is the centre of the circle of radius r cm which touches the above circles externally. IF ∆ ABC = 90 , write an equation in r and solve it.
​

Answer 1

Solution :-

In right triangle PQR, by Pythagoras theorem,

PQ² = PR² + RQ²

⇒ 17² = (x + 9)² + (x + 2)²

⇒ 298 = x² + 18² + 81 + x² + 4x + 4

⇒ 2x² + 22x - 204 = 0

⇒ x² + 11x - 102 = 0

⇒ x² + 17x - 6x - 102 = 0

⇒ x(x + 17) - 6(x + 17) = 0

⇒ (x - 6) (x + 17) = 0

⇒ x = 6, - 17 (x can't be negative)

⇒ x = 6 cm

Hence, the value of x is 6 cm.

Check the attachment for clear reference.