Question

If a point P has co-ordinates $(0,-2)$ and Q is any point on the circle, $x^2+y^2-5x-y+5=0$, then the maximum value of $(PQ{)}^2$ is :

• A. $\frac{25+\sqrt{6}}{2}$
• B. $14+5\sqrt{3}$
• C. $\frac{47+10\sqrt{6}}{2}$
• D. $8+5\sqrt{3}$

Answer 1

Answer: (B)

${\left(x-5/2\right)}^2\frac{-25}{4}+{\left(y-1/2\right)}^2-1/4+5=0$
$={\left(x-5/2\right)}^2+{\left(y-1/2\right)}^2=3/2$
on circle Q $=5/2+\sqrt{3/2}cosQ,\frac{1}{2}+\sqrt{3/2}sinQ$
$P{Q}^2={\left(\frac{5}{2}+\sqrt{3/2}cosQ\right)}^2+{\left(\frac{5}{2}+\sqrt{3/2}sinQ\right)}^2$
$P{Q}^2=\frac{25}{2}+\frac{3}{2}+5\sqrt{3/2}\left(cosQ+sinQ\right)$
$ma{n}^{mr}=14+5\sqrt{3/2}\left(\sqrt{2}\right)$
$=14+5\sqrt{3}$