'O' is the origin and Ak(xk, yk) where k = 1,2 are two points. If the circles are described on OA1 and OA2 as diameters, then the length of their common chord is equal to

Area of ΔOA1A2=12A1A2length of common chord⇒12x1y2−y1x2=12A1A2common chord⇒length of common chord=x1y2−x2y1A1A2

'O' is the origin and Ak(xk, yk) where k = 1,2 are two points. If the circles are described on OA1 and OA2 as diameters, then the length of their common chord is equal to

Area of ΔOA1A2=12A1A2length of common chord⇒12x1y2−y1x2=12A1A2common chord⇒length of common chord=x1y2−x2y1A1A2

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'O' is the origin and A_k(x_k, y_k) where k = 1,2 are two points. If the circles are described on OA₁ and OA₂ as diameters, then the length of their common chord is equal to

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$|x_{1} y_{2} - x_{2} y_{1}|$

$\frac{|x_{1} y_{2} - x_{2} y_{1}|}{A_{1} A_{2}}$

$\frac{1}{2} A_{1} A_{2}$

$\frac{1}{2} |x_{1} y_{2} - x_{2} y_{1}|$

answer is D.

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Detailed Solution

Area of ${ΔOA}_{1} A_{2} = \frac{1}{2} |A_{1} A_{2}| (length of common chord)$
$\begin{array}{l} \Rightarrow \frac{1}{2} |x_{1} y_{2} - y_{1} x_{2}| = \frac{1}{2} (A_{1} A_{2}) (c o m m o n c h o r d) \\ \Rightarrow (length of common chord) = \frac{|x_{1} y_{2} - x_{2} y_{1}|}{A_{1} A_{2}} \end{array}$