Question

Line ℓ touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following.

1. What is d(O, P) = ? Why ? 
2. If d(O, Q) = 8 cm, where does the point Q lie? 
3. If d(PQ) = 15 cm, How many locations of point R are line on line ℓ? At what distance will each of them be from point P?

Answer 1

i. seg OP is the radius of the circle.

∴ d(O, P) = 9 cm.

ii. Here, 8 cm < 9 cm

∴ d(O, Q) < d(O, P)

∴ d(O, Q) < radius

∴ Point Q lies in the interior of the circle.

iii. There can be two locations of point R on line `l`.

d(O, R) = 15 cm

Now, in ∆OPR, ∠OPR = 90°   ...[Tangent theorem]

∴ OR² = OP² + PR²    ...[Pythagoras theorem]

∴ 15² = 9² + PR²

∴ 225 = 81 + PR²

∴ PR² = 225 – 81

= 144

∴ PR = `sqrt(144)`

= 12 cm

∴ Each location of point R will be at a distance of 12 cm from point P.    ...[Taking square root of both sides]