If the line segment joining the points `P(x_1, y_1)a n d\ Q(x_2, y_2)` subtends an angle `alpha` at the origin O, prove that : `O PdotO Q\ cosalpha=x_1x_2+y_1y_2dot`

Options

A. `x_(1)x_(2)+y_(1)y_(2)`
B. `x_(1)y_(2)+x_(2)y_(1)`
C. `|x_(1)y_(2)-x_(2)y_(1)|`
D. none of these

Answer

A

Explanation

Using cosine formula in `DeltaPOQ` we have
`PQ^(2)=OP^(2)+OQ^(2)-2OPxxOQcos anglePOQ`
`rArr (x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2)x_(1)^(2)+y_(1)^(2)+x_(2)^(2)+y_(2)^(2)-2OPxxOQcos anglePOQ`
`rArr PO.OQ cos anglePOQ=x_(1)x_(2)+y_(1)y_(2)`