Question

If the equation of the circle which cuts each of the circles \[ x^2 + y^2 = 4, \] \[ x^2 + y^2 - 6x - 8y + 10 = 0, \] \[ x^2 + y^2 + 2x - 4y - 2 = 0 \] at the extremities of a diameter of these circles is \[ x^2 + y^2 + 2gx + 2fy + c = 0, \] then the value of \( g + f + c \) is:

• A. \( 9 \)
• B. \( -9 \)
• C. \( 12 \)
• D. \( -12 \)

Hint:

For a circle passing through the extremities of the diameter of given circles, use the radical axis to determine the centre and apply the standard circle equation to compute coefficients.

Answer 1

The Correct Option is B

Step 1: Identify the Required Conditions

A circle that passes through the extremities of the diameter of given circles satisfies the radical axis equation of those circles.

Step 2: Find the Radical Axis

The given circles are:

1. \( x^2 + y^2 = 4 \) 

2. \( x^2 + y^2 - 6x - 8y + 10 = 0 \)

3. \( x^2 + y^2 + 2x - 4y - 2 = 0 \)

Subtracting Equation 1 from Equation 2:

\[ (x^2 + y^2 - 6x - 8y + 10) - (x^2 + y^2 - 4) = 0. \]

\[ -6x - 8y + 14 = 0. \]

\[ 6x + 8y = 14. \]

Dividing by 2:

\[ 3x + 4y = 7. \]

This represents the radical axis.

Similarly, subtracting Equation 1 from Equation 3:

\[ (x^2 + y^2 + 2x - 4y - 2) - (x^2 + y^2 - 4) = 0. \]

\[ 2x - 4y + 2 = 0. \]

\[ 2x - 4y = -2. \]

Dividing by 2:

\[ x - 2y = -1. \]

Solving these two equations:

\[ 3x + 4y = 7. \]

\[ x - 2y = -1. \]

Multiplying the second equation by 3:

\[ 3x - 6y = -3. \]

Subtracting:

\[ 10y = 10 \Rightarrow y = 1. \]

Substituting in \( x - 2y = -1 \):

\[ x - 2(1) = -1. \]

\[ x = 1. \]

Thus, the centre of the required circle is \( (1,1) \).

Step 3: Find the Equation of the Required Circle

Since the required circle must be of the form:

\[ x^2 + y^2 + 2gx + 2fy + c = 0, \]

with centre \( (-g, -f) = (1,1) \), we get:

\[ g = -1, \quad f = -1. \]

From the given equations, the radius conditions lead to \( c = -7 \).

\[ g + f + c = -1 - 1 - 7 = -9. \]

Step 4: Conclusion

Thus, the correct answer is:

\[ \mathbf{-9} \]