Question

$\text{If the line segment joining the points}p(x_1,y_1)andQ(x_2,y_2)\text{subtends and angle}\alpha attheoriginO,provethat:OP.OQcos\alpha =x_1{x}_2+y_1{y}_2.$

Answer 1

$\text{It is given that O is the origin,}Then,O{Q}^2=x_2^2+y_2^{2}O{P}^2=x_1^2+y_1^{2}andP{Q}^2=(x_2-x_1{)}^2+(y_2-y_1{)}^{2}Usingcosineformulain\Delta OPQ,wehaveP{Q}^2=O{P}^2+O{Q}^2-2\left(OP\right)\left(OQ\right)cos\alpha \Rightarrow (x_2-x_1{)}^2+(y_2-y_1{)}^2=x_2^2+y_2^2+x_1^2+y_1^2-2\left(OP\right),\left(OQ\right)cos\alpha \Rightarrow x_2^2+x_1^2-2x_2{x}_1+y_2^2+y_1^2-2y_2{y}_1=x_2^2+y_2^2+x_1^2+y_1^2-2OP.OQcos\alpha \Rightarrow -2x_1{x}_2-2y_1{y}_2=-2OP.OQcos\alpha \Rightarrow x_1{x}_2+y_1{y}_2=OP.OQcos\alpha \Rightarrow OP.OQcos\alpha =x_1{x}_2+y_1{y}_{2}Hence,proved.$