Question

Consider the circles $S_1:x^2+y^2=4$ and $S_2:x^2+y^2-2x-4y+4=0.$ Which of the following statements is (are) CORRECT?

• A. Number of common tangents to these circles is $2.$
• B. If the power of a variable point $P$ with respect to these two circles is same, then $P$ moves on the line $x+2y-4=0$.
• C. Sum of the $y-$intercepts of both the circles is $6$.
• D. The circles $S_1$ and $S_2$ are orthogonal.

Answer 1

Answer: (A), (B), (D)

The circles $S_1$ and $S_2$ are orthogonal.

$S_1:x^2+y^2=4$
centre : $(0,0)$ ; radius $=2$
$S_2=x^2+y^2-2x-4y+4=0$
centre : $(1,2)$ ; radius $=1$

Distance between centres, $d=\sqrt{1^2+2^2}=\sqrt{5}$
$r_1+r_2=3,|r_1-r_2|=1$
$\therefore |r_1-r_2|<d<r_1+r_2$
$\therefore$ These two circles are intersecting.
$\therefore$ Number of common tangents is $2$.

$\to P(h,k)$ power of point $P$ is same with respect to these two circles.
$\therefore h^2+k^2-4=h^2+k^2-2h-4k+4$
$\Rightarrow -4=-2h-4k+4$
$\Rightarrow 2h+4k-8=0$
$\Rightarrow x+2y-4=0$

$\to y$-intercept of $S_1$ is $2\sqrt{4}=4$
$y$-intercept of $S_2$ is $2\sqrt{4-4}=0$
$\Rightarrow$ Sum of $y$-intercept $=4$

$\to 2(0+0)=-4+4$
$\Rightarrow 0=0$
$\therefore S_1$ and $S_2$ are orthogonal.