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(O, Q) = 15 cm, How many locations of point Q are line on line l? At what distance will each of them be from point P?
   

Answer 1

Radius of the circle = 9 cm

(1) It is given that line l is tangent to the circle at P.
∴ OP = 9 cm     ...(Radius of the circle)
d(O, P) = 9 cm

(2) d(O, Q) = 8 cm < Radius of the circle    
∴ Point Q lies in the interior of the circle.

(3) If d(OQ) = 15 cm, then there are two locations of point Q on the line l. One on the left of point P and one on the right of point P.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.
∴ ∠OPQ = 90º
In right ∆OPQ,
OQ² = OP² + PQ²
PQ = `sqrt("OQ"^2 - "OP"^2)`
PQ = `sqrt(15^2 - 9^2)`
PQ = `sqrt(225 - 81)`
PQ = `sqrt(144)`        ...[Taking the square root of both sides]
PQ = 12 cm

Thus, the two locations of the point Q on line l, which are at a distance of 12 cm from point P.