Question

$P(x_1,y_1)$ and $Q(x_2,y_2)$ are points in the plane such that $PQ$ subtends a right angle at the origin $O$, then

• A. $P(1,3),Q(-3,1)$
• B. $P(3,1),Q(1,-3)$
• C. $P(2,5),Q(5,-2)$
• D. $P(1,1),Q(-1,1)$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $P(3,1),Q(1,-3)$
B $P(2,5),Q(5,-2)$
C $P(1,1),Q(-1,1)$
D $P(1,3),Q(-3,1)$

As $PQ$ subtends a right angle at the origin $O,$ then.

$\Rightarrow {\left(OP\right)}^2+{\left(OQ\right)}^2={\left(PQ\right)}^2$
$\Rightarrow (x_1^2+y_1^2)+(x_2^2+y_2^2)=(x_1-x_2{)}^2+(y_1-y_2{)}^2$

$\Rightarrow (x_1{x}_2+y_1{y}_2)=0$

Then from aptions

$\left(A\right)1.(-3)+\left(3\right).\left(1\right)=-3+3=0$

$\left(B\right)3.1+1.(-3)=3-3=0$

$\left(C\right)2.5+\left(5\right).(-2)=10-10=0$

$\left(D\right)1.(-1)+\left(1\right).\left(1\right)=1+1=0$

Hence All the options are true.