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,R ) = 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) 9cm (2) In the interior of the circle . (3) 12 cm .

Explanation

(1) Radius of the circle is 9 cm ….(Given )
OP is the radius
`:. D ( O,P ) = 9 cm `
(2) `d(O,Q) = 8 cm `
`:' d (O,Q ) lt ` radius
`:.` point Q lies in the interior of the circle .
(3) If d(O,R ) = 15 cm , then there are two possible locations of point R on line l, one towards the left side of line OP and the other towards the right side of line OP. Let the location be R and `R_(1)`
In `Delta OPR `,
`/_OPR = 90^(@)`
`:.` by Pythagoras theorem,
`OR^(2) = OP^(2) + PR^(2)`
`:. 15^(2) = 9^(2) + PR^(2)`
`:. PR^(2) = 15^(2) - 9^(2)`
`:. PR^(2) = 225 - 81`
`:. PR^(2) = 144`
`:. PR = 12` cm ...(Taking square roots of both the sides )
Similarly `PR_(1) = 12 cm`
`:.` each loaction of point R will be at distance of 12 cm from point P .