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 x 2 - y 2 -10x-14y-51-0- then G-M- of p and q is equal to -

The correct option is D 2√( 11 ) Centre and radius of the given circle are C(5,7) and r=√(25+49+51)=5√(5) respectively.And let the given point ...

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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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2 \sqrt{5}

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13

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None of these

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Solution

The correct option is D $2 \sqrt{11}$
Centre and radius of the given circle are $C (5, 7)$ and $r = \sqrt{25 + 49 + 51} = 5 \sqrt{5}$ respectively.
And let the given point be $P (- 7, 2)$ .
Thus $C P = \sqrt{12^{2} + 5^{2}} = 13$
Therefore $p = C P + r = 13 + 5 \sqrt{5}, q = 13 - 5 \sqrt{5}$
$∴ G M = \sqrt{(13 + 5 \sqrt{5}) (13 - 5 \sqrt{5})}$ $= \sqrt{169 - 125} = \sqrt{44} = 2 \sqrt{11}$

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Similar questions

Q. 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

p

and

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is equal to

Q. Let

d_{1}, d_{2}

be the longest and shortest distance of a point

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from the curve

x^{2} + y^{2} - 10 x - 14 y - 51 = 0.

Then the geometric mean of

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from the curve whose equation is

x^{2} + y^{2} - 10 x - 14 y = 51

then Geometric mean of

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d_{2}

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d_{1}

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are the longest and the shortest distances of the point

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from the circle

x^{2} + y^{2} - 10 x - 14 y - 51 = 0,

then the value of

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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 G.M. of p and q is equal to :-

The correct option is D 2√11Centre and radius of the given circle are C(5,7) and r=√25+49+51=5√5 respectively.And let the given point be P(−7,2). Thus CP=√122+52=13Therefore p=CP+r=13+5√5,q=13−5√5 ∴GM=√(13+5√5)(13−5√5) =√169−125=√44=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 :-