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10-1.Circle and System of Circles
hard
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
$7$
(JEE MAIN-2021)
Solution
Centre of smallest circle is $\mathrm{A}$
Centre of largest circle is $\mathrm{B}$
$r_{2}=|C P-C A|=3 \sqrt{2}-3$
$r_{1}=C P+C B=3 \sqrt{2}+3$
$\frac{r_{1}}{r_{2}}=\frac{(3 \sqrt{2}+3)^{2}}{9}=(\sqrt{2}+1)^{2}=3+2 \sqrt{2}$
$a=3, b=2$
Standard 11
Mathematics
