Show the equation of a circle having the line segment joining $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ as diameter is $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$ .

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Solution

$m_{A P} = \frac{y - y_{1}}{x - x_{1}} m_{B P} = \frac{y - y_{2}}{x - x_{2}}$
Since $Δ A P B$ is a right angle (D in semicircle)
$m_{A P} \times m_{B P} = \frac{y - y_{1}}{x - x_{1}} \times \frac{y - y_{2}}{x - x_{2}} = - 1$
We get equation of the circle is
$(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0 x^{2} - x x_{2} - x_{1} x + x_{1} x_{2} + y^{2} - y y_{2} - y y_{1} + y_{1} y_{2} = 0 x^{2} + y^{2} - x (x_{2} + x_{1}) - y (y_{2} + y_{1}) + x_{1} x_{2} + y_{1} y_{2} = 0$

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