Question

The abscissae of A and B are the roots of the equation $x^{2} + 2ax - b^{2} = 0$ and their ordinates are the roots of the equation $y^{2} + 2by - q^{2} = 0.$ The equation of the circle with AB as diameter is

• A. $x^{2} + y^{2} + 2ax + 2by - b^{2} - q^{2} = 0$
• B. $x^{2} + y^{2} + 2ax + by - b^{2} - q^{2} = 0$
• C. $x^{2} + y^{2} + 2ax + 2by + b^{2} + q^{2} = 0$
• D. None of these

Answer 1

Answer: (A)

$x^{2} + y^{2} + 2ax + 2by - b^{2} - q^{2} = 0$

Answer 2

Let $x_{1},x_{2}$ and $y_{1},y_{2}$ be roots of $x^{2} + 2ax - b^{2} = 0$ and $y^{2} + 2by - q^{2} = 0$ respectively.

Then, $x_{1} + x_{2} = - 2a,x_{1}x_{2} = - b^{2}$ and

$y_{1} + y_{2} = - 2b,y_{1}y_{2} = - q^{2}$The equation of the circle with $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$ as the end points of diameter is

$\left( x - x _ { 1 } \right) \left( x - x _ { 2 } \right) + \left( y - y _ { 1 } \right) \left( y - y _ { 2 } \right) = 0$ $x^{2} + y^{2} - x(x_{1} + x_{2}) - y(y_{1} + y_{2}) + x_{1}x_{2} + y_{1}y_{2} = 0$; $x^{2} + y^{2} + 2ax + 2by - b^{2} - q^{2} = 0$