Question

A circle S given by \(x^2 + y^2 - 14x + 6y + 33 = 0\) cuts the X-axis at A and B (OB \(>\) OA). C is midpoint of AB. L is a line through C and having slope \((-1)\). If L is the diameter of a circle S' and also the radical axis of the circles S and S', then the equation of the circle S' is

• A. \(x^2 + y^2 - 17x + 3y + 54 = 0\)
• B. \(x^2 + y^2 + 17x - 3y - 54 = 0\)
• C. \(x^2 + y^2 - 17x + 3y + 51 = 0\)
• D. \(x^2 + y^2 - 3x + 17y - 51 = 0\)

Hint:

The radical axis of two circles \(S=0\) and \(S'=0\) (where coefficient of \(x^2, y^2\) is 1) is simply given by \(S - S' = 0\). If this axis is a specific line \(Ax+By+C=0\), you can compare coefficients to find unknown parameters.

Answer 1

The Correct Option is A

Step 1: Find points A and B:
The circle \(S\) cuts the x-axis, so set \(y=0\) in \(S\): \[ x^2 - 14x + 33 = 0 \] \[ (x-3)(x-11) = 0 \implies x = 3, 11 \] So, \(A(3, 0)\) and \(B(11, 0)\).
Step 2: Find Midpoint C and Equation of Line L:
Midpoint \(C\) of \(AB\) is \((\frac{3+11}{2}, 0) = (7, 0)\). Line \(L\) passes through \(C(7,0)\) with slope \(m = -1\). Equation of \(L\): \(y - 0 = -1(x - 7) \implies x + y - 7 = 0\).
Step 3: Use Properties of S':
Let the circle \(S'\) be \(x^2 + y^2 + 2gx + 2fy + c = 0\). 1. L is the diameter of S':
The center \((-g, -f)\) lies on \(L\). \[ (-g) + (-f) - 7 = 0 \implies g + f = -7 \quad \dots(1) \] 2. L is the radical axis of S and S':
The radical axis equation is \(S - S' = 0\). \(S: x^2 + y^2 - 14x + 6y + 33 = 0\) \(S - S': (-14 - 2g)x + (6 - 2f)y + (33 - c) = 0\). This line must be identical to \(L: x + y - 7 = 0\). Comparing coefficients: \[ \frac{-14 - 2g}{1} = \frac{6 - 2f}{1} = \frac{33 - c}{-7} \]
Step 4: Solve for g, f, and c:
From first equality: \(-14 - 2g = 6 - 2f\) \(2f - 2g = 20 \implies f - g = 10 \quad \dots(2)\) Solving (1) and (2): Adding equations: \(2f = 3 \implies f = \frac{3}{2}\). Subtracting: \(2g = -17 \implies g = -\frac{17}{2}\). From the constant term ratio: \[ 6 - 2f = \frac{33 - c}{-7} \] Substitute \(f = 3/2\): \[ 6 - 3 = \frac{33 - c}{-7} \implies 3 = \frac{33 - c}{-7} \] \[ -21 = 33 - c \implies c = 54 \]
Step 5: Write Equation of S':
Substitute \(g, f, c\) into the general equation: \[ x^2 + y^2 + 2(-\frac{17}{2})x + 2(\frac{3}{2})y + 54 = 0 \] \[ x^2 + y^2 - 17x + 3y + 54 = 0 \]