Question

The perpendicular from origin on the line joining the points $P(\mathrm{acos}\alpha ,\mathrm{asin}\alpha ),Q(\mathrm{acos}\beta ,\mathrm{asin}\beta )$ divides $PQ$ in the ratio

• A. $5:3$
• B. $2:7$
• C. $1:1$
• D. $3:4$

Answer 1

The correct option is C $1:1$

As $O$ is the center and $\perp r$ from the origin to the line $PQ$

$\left(PDO\right)={90}^o$

$OA=\sqrt{a{\cos }^2\alpha +a^2{\sin }^2\alpha }=\sqrt{a^2}=a$

$OB=\sqrt{a^2{\cos }^2\beta +a^2{\sin }^2\beta }=\sqrt{a^2}=a$

$\therefore OA=OB=a$

$\therefore △OAB$ is isosceles.

Hence $OD$ bisect $PQ$

$\Rightarrow PD=DQ$

$\therefore PD:DQ=1:1$