If O be the origin and if co - ordinates of two any points Q1andQ2be(x1,y1)and(x2,y2)

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Solution

From triangles $O Q_{1} Q_{2}$ by applying consine formula we , get
$Q_{1} Q_{2}^{2} = O Q_{1}^{2} + O Q_{2}^{2} - 2 O Q_{2} \cdot O Q_{2} c o s Q_{1} O Q_{2}$
or $(x_{1} - x_{2})^{2} (y_{1} - y_{2})^{2}$
$= x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2} - 2 O Q_{1} \cdot O Q_{2} \cdot c o s θ$
or $x_{1}^{2} + x_{2}^{2} - 2 x_{1} x_{2} + y_{1}^{2} + y_{2}^{2} - 2 y_{1} y_{2}$
$= x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2} - 2 O Q_{1} \cdot O Q_{2} c o s θ$
or $x_{1} x_{2} + y_{1} y_{2} = O Q_{1} \cdot O Q_{2} c o s Q_{1} O Q_{2}$

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