Question

If the equation of the circle passing through the points (-1,0), (-1,1), (1,1) is \( ax^2+ay^2+2gx+2fy-2=0 \) then \( a = \)

• A. 1
• B. -1
• C. 2
• D. -2

Hint:

Notice the symmetry: Points (-1,1) and (1,1) imply the center's x-coordinate is 0 (so $g=0$). Points (-1,0) and (-1,1) imply the center's y-coordinate is 0.5. Since $g=0$, substitution becomes trivial.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Substitute the coordinates of the given points into the equation of the circle to obtain simultaneous linear equations in \( a, g, f \), and then solve for \( a \).
Step 2: Detailed Explanation:
Equation: \( ax^2 + ay^2 + 2gx + 2fy - 2 = 0 \). 1. Point \( (-1, 0) \): \( a(1) + 0 + 2g(-1) + 0 - 2 = 0 \implies a - 2g - 2 = 0 \implies 2g = a - 2 \). (Eq 1) 2. Point \( (-1, 1) \): \( a(1) + a(1) + 2g(-1) + 2f(1) - 2 = 0 \implies 2a - 2g + 2f - 2 = 0 \). (Eq 2) 3. Point \( (1, 1) \): \( a(1) + a(1) + 2g(1) + 2f(1) - 2 = 0 \implies 2a + 2g + 2f - 2 = 0 \). (Eq 3) Subtract Eq 2 from Eq 3: \( (2a + 2g + 2f - 2) - (2a - 2g + 2f - 2) = 0 \) \( 4g = 0 \implies g = 0 \). Substitute \( g = 0 \) into Eq 1: \( a - 0 - 2 = 0 \implies a = 2 \).
Step 3: Final Answer:
The value of \( a \) is 2.