Question

Let the coordinates of points A and B satisfy:
- $ x^2 + 2x - a^2 = 0 $
- $ y^2 + 4y - b^2 = 0 $ Find the equation of the circle having AB as diameter.

• A. \( x^2 + y^2 + 2x + 4y - a^2 - b^2 = 0 \)
• B. \( x^2 + y^2 + 2x + 4y + a^2 + b^2 = 0 \)
• C. \( x^2 + y^2 - 2x - 4y - a^2 - b^2 = 0 \)
• D. \( x^2 + y^2 - 2x - 4y + a^2 + b^2 = 0 \)

Hint:

The equation of a circle with diameter AB is derived by expanding \( (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \).

Answer 1

The Correct Option is A

The roots of the equations: - \( x^2 + 2x - a^2 = 0 \Rightarrow x = -1 \pm \sqrt{1 + a^2} \) - \( y^2 + 4y - b^2 = 0 \Rightarrow y = -2 \pm \sqrt{4 + b^2} \) So points A and B are: - \( A = (x_1, y_1), B = (x_2, y_2) \) where: \[ x_1 + x_2 = -2,\quad y_1 + y_2 = -4,\quad x_1 x_2 = -a^2,\quad y_1 y_2 = -b^2 \] Midpoint \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = (-1, -2) \) Using formula for circle with diameter AB: \[ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 \Rightarrow x^2 - (x_1 + x_2)x + x_1 x_2 + y^2 - (y_1 + y_2)y + y_1 y_2 = 0 \] Substitute: - \( x_1 + x_2 = -2 \) - \( x_1 x_2 = -a^2 \) - \( y_1 + y_2 = -4 \) - \( y_1 y_2 = -b^2 \) \[ x^2 + 2x - a^2 + y^2 + 4y - b^2 = 0 \Rightarrow x^2 + y^2 + 2x + 4y - a^2 - b^2 = 0 \] \[ \boxed{ x^2 + y^2 + 2x + 4y - a^2 - b^2 = 0 } \]