Line l touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following: d(O, P), position of Q if d(O, Q) = 8 cm, and locations of point R on line l.

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?

(In part (3) the intended condition is d(O, R) = 15 cm for a point R on line \(l\).)

Doubtly Asked 24 Oct 2026 Save WhatsApp Report

1 answer

Answer: (1) 9 cm (2) in the interior of the circle (3) two locations, each 12 cm from P

(1) d(O, P) = 9 cm, because OP is a radius (P is the point of contact). Also OP ⊥ \(l\), so 9 cm is the distance of the centre from line \(l\).

(2) d(O, Q) = 8 cm is less than the radius 9 cm, so Q lies in the interior of the circle.

(3) For a point R on \(l\) with OR = 15 cm, \(\triangle OPR\) is right angled at P:

\[PR^2 = OR^2 - OP^2 = 225 - 81 = 144\] \[PR = 12\ \text{cm}\]

Such a point can be on either side of P on line \(l\), so there are two locations of R, each 12 cm from P.

Doubtly AI · 24 Oct 2026 AI-verified · Oct 2026 Report

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