Let r_1 and r_2 be the radii of the largest and smallest circles, respectively, which pass…

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 have their centres on the circumference of the circle $x^2+y^2+2x+4y-4=0$. If $\dfrac{r_1}{r_2}=a+b\sqrt2$, then $a+b$ is equal to

Accepted Answer

Centre of smallest circle is A

Centre of largest circle is B

$${r_2} = |CP - CA| = 3\sqrt 2  - 3$$

$${r_1} = CP - CB = 3\sqrt 2  + 3$$

$${{{r_1}} \over {{r_2}}} = {{3\sqrt 2  + 3} \over {3\sqrt 2  - 3}} = {{{{(3\sqrt 2  + 3)}^2}} \over 9} = {(\sqrt 2  + 1)^2} = 3 + 2\sqrt 2 $$

a = 3, b = 2

Solutions · 1