Question

Consider the circles
$S_1:x^2+y^2-12y+35=0$
$S_2:x^2+y^2-4x-4y+4=0$
$S_1:x^2+y^2-9x=0$
Which of the following statements is correct?

• A. The centres of these three circles form a right triangle.
• B. Point $(4,1)$ lies outside $S_1$ but inside $S_2$ and $S_3$.
• C. The length of the chord intercepted on the line $y=x$ by $S_3$ is $3\sqrt{2}.$
• D. Equation of the tangent on circle $S_3=0$ at the origin is $y=0$

Answer 1

The correct option is C The length of the chord intercepted on the line $y=x$ by $S_3$ is $3\sqrt{2}.$

$C_1(0,6),C_2(2,2),C_3(3,0)$ are centres of $S_1,S_2,S_3$ respectively which are collinear on line $2x+y=6$

For point $(4,1)$
$S_1>0$ and $S_2>0$ and $S_3<0$
$\therefore$ Point $(4,1)$ lies inside $S_3$ and outside $S_1$ and $S_2$

$p=\left|\frac{3}{\sqrt{2}}\right|$
Now, $2l=2\sqrt{9-\frac{9}{2}}=3\sqrt{2}$


Tangent at $(0,0)$ on $S_3=0$ is $x=0$.