Question

Let the circle \( S: x^2 + y^2 + 2gx + 2fy + c = 0 \) cut the circles \( x^2 + y^2 - 2x + 2y - 2 = 0 \) and \( x^2 + y^2 + 4x - 6y + 9 = 0 \) orthogonally. If the centre of the circle \( S = 0 \) lies on the line \( 2x + 3y - 2 = 0 \), then \( 2g + f = \)

• A. \( c \)
• B. \( c + f \)
• C. \( 2g - c \)
• D. \( c - f \)

Hint:

When dealing with the orthogonality of two circles, remember to use the orthogonality condition: \( 2g_1g_2 + 2f_1f_2 + c_1c_2 = 0 \) and apply the given geometric conditions.

Answer 1

The Correct Option is D

We are given three circles, and we know that the circle \( S \) cuts the other two circles orthogonally. This means the condition of orthogonality for two circles can be applied. For the equation of circle \( S: x^2 + y^2 + 2gx + 2fy + c = 0 \), and the other two given circles, we use the condition for orthogonality: \[ 2g_1g_2 + 2f_1f_2 + c_1c_2 = 0 \] Since the center of circle \( S \) lies on the line \( 2x + 3y - 2 = 0 \), we substitute this condition and solve for \( 2g + f \). After simplifying the equations, we find that the correct value of \( 2g + f \) is \( c - f \). Therefore, the correct answer is \( c - f \).