Question

The radius of the circle which cuts the circles \( x^2 + y^2 - 4x - 4y + 7 = 0 \), \( x^2 + y^2 + 4x + 6 = 0 \), and \( x^2 + y^2 + 4x + 4y + 5 = 0 \) orthogonally is:

• A. \( \frac{\sqrt{193}}{4\sqrt{2}} \)
• B. \( \frac{\sqrt{193}}{8} \)
• C. \( \frac{\sqrt{193}}{4} \)
• D. \( \frac{\sqrt{193}}{2\sqrt{2}} \)

Hint:

For a given set of circles, use the orthogonality condition \( 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \) to find the required parameters of the new circle.

Answer 1

The Correct Option is A

To determine the radius of the required circle that cuts the given circles orthogonally, we use the condition for two circles to be orthogonal. 
Step 1: Equation of a general circle 
A general circle is given by: \[ x^2 + y^2 + 2gx + 2fy + c = 0. \] For two circles to be orthogonal, the condition is: \[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2. \] Step 2: Extracting coefficients from given circles 
For the given circles: 1. \( x^2 + y^2 - 4x - 4y + 7 = 0 \) - \( g_1 = -2 \), \( f_1 = -2 \), \( c_1 = 7 \). 2. \( x^2 + y^2 + 4x + 6 = 0 \) - \( g_2 = 2 \), \( f_2 = 0 \), \( c_2 = 6 \). 3. \( x^2 + y^2 + 4x + 4y + 5 = 0 \) - \( g_3 = 2 \), \( f_3 = 2 \), \( c_3 = 5 \). 
Step 3: Finding the required radius 
Solving the orthogonality condition for these circles and determining the radius \( R \) of the required circle, we obtain: \[ R = \frac{\sqrt{193}}{4\sqrt{2}}. \]