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Multiplication with Vectors

If d1 d...

Question

If $d_{1}$ & $d_{2}$ are the longest and shortest distances of $(- 7, 2)$ from the curve whose equation is $x^{2} + y^{2} - 10 x - 14 y = 51$ then Geometric mean of $d_{1}$ & $d_{2}$ is?

A

\sqrt{11}

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B

7

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C

2

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D

2 \sqrt{11}

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Solution

The correct option is D $2 \sqrt{11}$
$x^{2} + y^{2} - 10 x - 14 y - 51 = 0$ $P (- 7, 2)$
Center $(5, 7)$ Radius = $5 \sqrt{5}$
Longer distance $d_{1} = \sqrt{(7 + 5)^{2} + (7 - 2)^{2}} + 5 \sqrt{5}$
$= 13 + 5 \sqrt{5}$
Short distance $d_{2} = 13 - 5 \sqrt{5}$
$G M = \sqrt{(13 + 5 \sqrt{5}) (13 - 5 \sqrt{5})} = \sqrt{169 - 125}$
$= \sqrt{44}$
$= 2 \sqrt{11}$

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