If p and q be the longest and the shortest distance respectively of the point $(-7,2)$ from any point…
If p and q be the longest and the shortest distance respectively of the point $(-7,2)$ from any point $(\alpha, \beta)$ on the curve whose equation is $x^2+y^2-10 x-14 y-51=0$, then G.M. of p
$2 \sqrt{11}$
$5 \sqrt{5}$
$13$
$11$
Solution
The centre $C$ of the circle $=(5,7)$ and the radius
$=\sqrt{5^2+7^2+51}=5 \sqrt{5}$
$\mathrm{PC}=\sqrt{12^2+5^2}=13 \Rightarrow q=P A=13-5 \sqrt{5}$
and $p=P B=13+5 \sqrt{3}$
$\therefore$ G.M. of $p$ and $q$
$=\sqrt{p q}=\sqrt{(13-5 \sqrt{5})(13+5 \sqrt{5})}$
$=\sqrt{169-125}=2 \sqrt{11}$.
$\therefore$ G.M. of $p$ and $q$
$=\sqrt{p q}=\sqrt{(13-5 \sqrt{5})(13+5 \sqrt{5})}$
$=\sqrt{169-125}=2 \sqrt{11}$.