Question

If the centre $(\alpha, \beta)$ of a circle cutting the circles $x^2+y^2-2y-3=0$ and $x^2+y^2+4x+3=0$ orthogonally lies on the line $2x-3y+4=0$, then $2\alpha+\beta=$

• A. 3
• B. -3
• C. 0
• D. 1

Hint:

The locus of the center of a circle that cuts two given circles orthogonally is their radical axis. This is a very useful property that can simplify problems significantly. Finding the radical axis by simply subtracting the equations of the circles (in the standard form with leading coefficients of 1) is a quick and effective method.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
A circle that cuts two given circles orthogonally must have its center lying on the radical axis of the two given circles. We first find the equation of the radical axis. Then, since the center $(\alpha, \beta)$ is also given to lie on another line, it must be the intersection of the radical axis and this given line. However, we may not need to find the center explicitly.
Step 2: Key Formula or Approach
1. The condition for two circles $x^2+y^2+2g_1x+2f_1y+c_1=0$ and $x^2+y^2+2g_2x+2f_2y+c_2=0$ to be orthogonal is $2g_1g_2 + 2f_1f_2 = c_1+c_2$. 2. The radical axis of two circles $S_1=0$ and $S_2=0$ is given by the equation $S_1 - S_2 = 0$. 3. The center of a circle that cuts two given circles orthogonally must lie on their radical axis.
Step 3: Detailed Explanation
Let the two given circles be: $S_1: x^2+y^2-2y-3=0$ $S_2: x^2+y^2+4x+3=0$ The center of the required circle is $(\alpha, \beta)$. This circle cuts both $S_1$ and $S_2$ orthogonally. Therefore, its center $(\alpha, \beta)$ must lie on the radical axis of $S_1$ and $S_2$. The equation of the radical axis is $S_1 - S_2 = 0$: \[ (x^2+y^2-2y-3) - (x^2+y^2+4x+3) = 0 \] \[ x^2+y^2-2y-3 - x^2-y^2-4x-3 = 0 \] \[ -4x - 2y - 6 = 0 \] Dividing by -2, we get the equation of the radical axis: \[ 2x + y + 3 = 0 \] Since the center $(\alpha, \beta)$ lies on this radical axis, its coordinates must satisfy this equation: \[ 2\alpha + \beta + 3 = 0 \] \[ 2\alpha + \beta = -3 \] The problem gives another piece of information that the center lies on the line $2x-3y+4=0$, which would allow us to find the exact coordinates of the center. However, the question only asks for the value of the expression $2\alpha+\beta$. We have already found this value from the radical axis property. Step 4: Final Answer
The center $(\alpha, \beta)$ of the circle must lie on the radical axis of the two given circles. The equation of the radical axis is $2x+y+3=0$. Therefore, $2\alpha+\beta+3=0$, which implies $2\alpha+\beta = -3$.