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(OQ) = 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,

$$\begin{gathered} {\mathrm{OQ}}^2={\mathrm{OP}}^2+{\mathrm{PQ}}^2 \\ \Rightarrow \mathrm{PQ}=\sqrt{{\mathrm{OQ}}^2-{\mathrm{OP}}^2} \\ \Rightarrow \mathrm{PQ}=\sqrt{{\left(15\right)}^2-{\left(9\right)}^2} \\ \Rightarrow \mathrm{PQ}=\sqrt{225-81} \\ \Rightarrow \mathrm{PQ}=\sqrt{144}=12\mathrm{cm} \end{gathered}$$
Thus, the two locations of the point Q on line l, which are at a distance of 12 cm from point P.