Question

If the circle $S=0$ intersect the three circles $S_1 = x^2+y^2+4x-7=0$, $S_2 = x^2+y^2+y=0$ and $S_3 = x^2+y^2+\frac{3}{2}x+\frac{5}{2}y-\frac{9}{2}=0$ orthogonally, then the radical axis of $S=0$ and $S_1=0$ is

• A. $4x-y-7=0$
• B. $x+y-3=0$
• C. $4x+y-3=0$
• D. $x-y-2=0$

Hint:

The radical axis of two circles $S_1=0$ and $S_2=0$ is the line $S_1-S_2=0$. The center of a circle S which is orthogonal to three given circles is the radical center of those three circles (the intersection of their radical axes).

Answer 1

The Correct Option is C

Step 1: Understand the property of the center of the orthogonal circle.
A circle S that intersects three circles $S_1, S_2, S_3$ orthogonally has its center at the radical center of $S_1, S_2, S_3$. The radical center is the point of intersection of the radical axes of pairs of the three circles.

Step 2: Find the radical axis of $S_1$ and $S_2$.
The equation of the radical axis is $S_1 - S_2 = 0$. \[ (x^2+y^2+4x-7) - (x^2+y^2+y) = 0 \implies 4x-y-7=0. (\text{RA}_{12}) \]

Step 3: Find the radical axis of $S_2$ and $S_3$.
The equation is $S_2 - S_3 = 0$. \[ (x^2+y^2+y) - \left(x^2+y^2+\frac{3}{2}x+\frac{5}{2}y-\frac{9}{2}\right) = 0. \] \[ y - \frac{3}{2}x - \frac{5}{2}y + \frac{9}{2} = 0. \] Multiplying by 2 gives $2y - 3x - 5y + 9 = 0 \implies -3x-3y+9=0 \implies x+y-3=0. (\text{RA}_{23})$

Step 4: Find the radical center (center of S).
We solve the system of equations for the radical axes. 1) $4x-y=7$ 2) $x+y=3$ Adding them gives $5x=10 \implies x=2$. Substituting into (2), $2+y=3 \implies y=1$. The center of circle S is $(2,1)$. Let the equation of S be $x^2+y^2-4x-2y+c=0$.

Step 5: Find the constant c for circle S.
S cuts $S_1$ orthogonally. The condition is $2g_S g_1 + 2f_S f_1 = c_S + c_1$. For S: $g_S=-2, f_S=-1, c_S=c$. For $S_1$: $g_1=2, f_1=0, c_1=-7$. \[ 2(-2)(2) + 2(-1)(0) = c + (-7) \implies -8 = c - 7 \implies c = -1. \] So, the equation of circle S is $x^2+y^2-4x-2y-1=0$.

Step 6: Find the radical axis of S and $S_1$.
The required radical axis is given by the equation $S - S_1 = 0$. \[ (x^2+y^2-4x-2y-1) - (x^2+y^2+4x-7) = 0. \] \[ -4x - 2y - 1 - 4x + 7 = 0. \] \[ -8x - 2y + 6 = 0. \] Dividing by -2, we get $4x+y-3=0$. This is the required radical axis.