If p and q be the longest distance and the shortest distance respectively of the point -7- 2 from any point - - - on the curve whose equation is x2-y2-10x-14y-51-0 then GM of p and q is equal to

The correct option is A 2√(11) Centre and the radius of the given circle are C(5,7) and r=√(25+49+51)=5√(5) respectively.Let P=( 7,2) be ...

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Byju's Answer

Standard XII

Mathematics

Perpendicular Distance of a Point from a Line

If p and ...

Question

If $p$ and $q$ be the longest distance and the shortest distance respectively of the point $(- 7, 2)$ from any point $(α, β)$ on the curve whose equation is $x^{2} + y^{2} - 10 x - 14 y - 51 = 0$ then $G M$ of $p$ and $q$ is equal to

2 \sqrt{11}

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5 \sqrt{5}

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13

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none of these

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Solution

The correct option is A $2 \sqrt{11}$
Centre and the radius of the given circle are $C (5, 7)$ and $r = \sqrt{25 + 49 + 51} = 5 \sqrt{5}$ respectively.
Let $P = (- 7, 2)$ be the given point.
Now $C P = \sqrt{12^{2} + 5^{2}} = 13$
Thus $p = C P + r = 13 + 5 \sqrt{5}$ and $q = C P - r = 13 - 5 \sqrt{5}$
Hence G.M of p and q is $= \sqrt{p q} = \sqrt{13^{2} - 125} = 2 \sqrt{11}$

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If p and q be the longest distance and the shortest distance respectively of the point (−7,2) from any point (α,β) on the curve whose equation is x2+y2−10x−14y−51=0 then GM of p and q is equal to

The correct option is A 2√11Centre and the radius of the given circle are C(5,7) and r=√25+49+51=5√5 respectively.Let P=(−7,2) be the given point.Now CP=√122+52=13Thus p=CP+r=13+5√5 and q=CP−r=13−5√5Hence G.M of p and q is =√pq=√132−125=2√11

If $p$ and $q$ be the longest distance and the shortest distance respectively of the point $(- 7, 2)$ from any point $(α, β)$ on the curve whose equation is $x^{2} + y^{2} - 10 x - 14 y - 51 = 0$ then $G M$ of $p$ and $q$ is equal to