Question

Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle $x^{2}+y^{2}+2 x+4 y-4=0$. If $\frac{r_{1}}{r_{2}}=a+b \sqrt{2}$, then $a+b$ is equal to :

• A. 3
• B. 11
• C. 5
• D. 7

Answer 1

Answer: (C)

Centre of smallest circle is $A$
Centre of largest circle is $B$
$r_{2}=|C P-C A|=3 \sqrt{2}-3 $
$r_{1}=C P+C B=3 \sqrt{2}+3$
$\frac{r_{1}}{r_{2}}=\frac{3 \sqrt{2}+3}{3 \sqrt{2}-3}=\frac{(3 \sqrt{2}+3)^{2}}{9}$
$=(\sqrt{2}+1)^{2}=3+2 \sqrt{2} $
$a=3, b=2$