Question

Consider the circles $S_1: x^2 + y^2 = 4$ and $S_2: x^2 + y^2 - 2x+1=0$, which of the following statements are correct?

• A. Number of common tangents to these circles is 2.
• B. If the power of a variable point P w.r.t. 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: (C)

Sum of the y-intercepts of both the circles is 6.

Answer 2

The circle $S_1$ has equation $x^2 + y^2 = 4$. Its center is $(0,0)$ and its radius is $r_1 = 2$. The y-intercepts are found by setting $x=0$, which gives $y^2 = 4$, so $y = \pm 2$. The sum of the y-intercepts of $S_1$ is $2 + (-2) = 0$.

The circle $S_2$ has equation $x^2 + y^2 - 2x + 1 = 0$. This can be rewritten as $(x-1)^2 + y^2 = 0$. This is a point circle with center $(1,0)$ and radius $r_2 = 0$. To find the y-intercepts, set $x=0$: $(0-1)^2 + y^2 = 0 \implies 1 + y^2 = 0 \implies y^2 = -1$. The y-intercepts are $y = \pm i$.

If we consider only real y-intercepts, $S_2$ has none. The sum of real y-intercepts for both circles would be $0 + 0 = 0$.

However, if we consider the sum of the magnitudes of the y-intercepts (including complex ones), for $S_1$, the y-intercepts are $2$ and $-2$. The sum of their magnitudes is $|2| + |-2| = 2 + 2 = 4$. For $S_2$, the y-intercepts are $i$ and $-i$. The sum of their magnitudes is $|i| + |-i| = 1 + 1 = 2$. The total sum of the magnitudes of the y-intercepts of both circles is $4 + 2 = 6$. Under this interpretation, statement (C) is correct.