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Question

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


Solution

From triangles $$OQ_1 \, Q_2 $$ by applying consine  formula we , get 
$$Q_1 Q_2^2 \, = \, OQ_1^2 \, + \, OQ_2^2 \, - \, 2OQ_2  \cdot \, OQ_2 \, cos \, Q_1OQ_2$$
or $$(x_1 - x_2)^2 \, (y_1 - y_2)^2$$
$$ = \, x_1^2 \, + \, y_1^2 \, + \, x_2^2 \, + \, y_2^2 \, - \, 2OQ_1 \, \cdot \, OQ_2\, \cdot \, cos \theta$$
or $$x_1^2 \, + \, x_2^2 \, - \, 2x_1 \, x_2 \, + \, y_1^2 \, + \, y_2^2 \, - \, 2y_1y_2$$
$$= \, x_1^2 \, + \,y_1^2 \, + x_2^2 \, + \, y_2^2 \, - \, 2OQ_1 \, \cdot \, OQ_2 \, cos \, \theta$$
or $$x_1x_2 \, + \, y_1y_2 \, = \, OQ_1 \, \cdot \, OQ_2 \, cos \, Q_1 \, OQ_2$$
1035344_1006404_ans_7eeb717b53f94063b57e3527db917b0a.png

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