$A (x_{1}, y_{1}), B (x_{2}, y_{2})$ are two given points. Circles are drawn on OA and OB as diameters where O is origin . Prove that the length of the common chord is $\frac{x_{1} y_{2} - x_{2} y_{1}}{A B}$

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Solution

Let OC be the common chord of the circles drawn on OA and OB as diameters. Then angle in a semi-circle being a right angle
$∠ O C A = ∠ O C B = 90^{\circ}$
$∴$ ACB is a straight line
$Δ O A B = \frac{1}{2} (x_{1} y_{2} - x_{2} y_{1}) = \frac{1}{2} O c \cdot A B$
$∴$ OC = length of c.c. = $\frac{x_{1} y_{2} - x_{2} y_{1}}{A B}$

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