The co-ordinates of two points $P$ and $Q$ are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ and $O$ is the origin. If circles be described on OP and OQ as diameters then length of their common chord

\frac{| x_{1} y_{2} + x_{2} y_{1} |}{P Q}

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\frac{| x_{1} y_{2} - x_{2} y_{1} |}{P Q}

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\frac{| x_{1} x_{2} + y_{1} y_{2} |}{P Q}

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\frac{| x_{1} x_{2} - y_{1} y_{2} |}{P Q}

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Solution

The correct option is A $\frac{| x_{1} y_{2} + x_{2} y_{1} |}{P Q}$
Equation of the circles are $x^{2} + y^{2} - x_{1} x - y_{1} y = 0$
and $x^{2} + y^{2} - x_{2} x - y_{2} y = 0$
Equation of the common chord is $y = x,$ for the point of intersection
${2 x}^{2} - (x_{1} + y_{1}) x = 0, {2 x}^{2} - (x_{2} + y_{2}) x = 0$
$∴$ length of chord $= \frac{| x_{1} y_{2} + x_{2} y_{1} |}{P Q}$

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The co-ordinates of two points P and Q are (x1,y1) and (x2,y2) and O is the origin. If circles be described on OP and OQ as diameters then length of their common chord

The correct option is A |x1y2+x2y1|PQEquation of the circles are x2+y2−x1x−y1y=0and x2+y2−x2x−y2y=0Equation of the common chord is y=x, for the point of intersection 2x2−(x1+y1)x=0,2x2−(x2+y2)x=0∴ length of chord =|x1y2+x2y1|PQ

The co-ordinates of two points $P$ and $Q$ are $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ and $O$ is the origin. If circles be described on OP and OQ as diameters then length of their common chord