First slide

Definition of a circle

Question

The equation of the circle whose one diameter is PQ, where the ordinates of P, Q are the roots of the equation $x^{2} + 2 x - 3 = 0$ and the abscissae are the roots of the equation $y^{2} + 4 y - 12 = 0$ ,is

Moderate

A

$x^{2} + y^{2} + 2 x + 4 y - 15 = 0$
B

$x^{2} + y^{2} - 4 x - 2 y - 15 = 0$
C

$x^{2} + y^{2} + 4 x + 2 y - 15 = 0$
D

none of these

Solution

Let $P (x_{1}, y_{1}) and Q (x_{2}, y_{2})$ be the end points of diameter PQ. Then
$x_{1} + x_{2} = - 2, x_{1} x_{2} = - 3, y_{1} + y_{2} = - 4 and y_{1} y_{/ 2} = - 12$
The equation of the circle having $P Q$ as a diameter is
$\begin{array}{l} x^{2} + y^{2} - x (x_{1} + x_{2}) - y (y_{1} + y_{2}) + x_{1} x_{2} + y_{1} y_{2} = 0 \\ x^{2} + y^{2} + 2 x + 4 y - 3 - 12 = 0 \\ x^{2} + y^{2} + 2 x + 4 y - 15 = 0 \end{array}$

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