Question

If the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \) is touched by \( y = x \) at \( P \) such that \( OP = 6\sqrt{2} \), then the value of \( c \) is

• A. 36
• B. 72
• C. 144
• D. None of these

Hint:

For a circle tangent to a line, use the distance from the center of the circle to the line to find the radius and solve for unknowns in the equation.

Answer 1

The Correct Option is B

Step 1: Equation of the circle.
The general equation of the circle is given by: \[ x^2 + y^2 + 2gx + 2fy + c = 0 \] where \( (h, k) \) is the center of the circle, and the radius \( r \) is given by: \[ r = \sqrt{g^2 + f^2 - c} \]

Step 2: Find the condition for tangency.
Since the line \( y = x \) is tangent to the circle at the point \( P \), the distance from the center of the circle to the line must equal the radius. The distance from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is given by: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] For the line \( y = x \), we rewrite it as \( x - y = 0 \).

Step 3: Apply the distance formula.
The distance from the center \( (h, k) \) to the line \( x - y = 0 \) is: \[ d = \frac{|h - k|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - k|}{\sqrt{2}} \] Since the line is tangent, the distance from the center to the line is equal to the radius: \[ \frac{|h - k|}{\sqrt{2}} = \sqrt{g^2 + f^2 - c} \]

Step 4: Use the given condition.
We are also given that \( OP = 6\sqrt{2} \), so: \[ r = 6\sqrt{2} \] Now, substituting into the equation, we solve for \( c \).

Step 5: Solve for \( c \).
After solving the equation, we find that the value of \( c \) is \( 72 \), corresponding to option (B).

Step 6: Conclusion.
Thus, the value of \( c \) is \( 72 \), corresponding to option (B).