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 $x^2+y^2+2x+4y-4=0.$ If
$\frac{r_1}{r_2}=a+b\sqrt{2,}$ then $a+b$ is equal to:

• A. $3$
• B. $7$
• C. $11$
• D. $5$

Answer 1

The correct option is D $5$

Given circle equation is $C\equiv {(x+1)}^2+{(y+2)}^2=9$
Distance between $(–1,–2)$ and $(–4,1)$ is $\sqrt{3^2+3^2}=\sqrt{18}$
Maximum radius of required circle $r_1=\sqrt{18}+3$
Minimum radius of required circle $r_2=\sqrt{18}-3$
$\frac{r_1}{r_2}=\frac{3\sqrt{2}+3}{3\sqrt{2}-3}=\frac{\sqrt{2}+1}{\sqrt{2}-1}=\frac{{(\sqrt{2}+1)}^2}{1}=3+2\sqrt{2}$