Question

Consider a $△PQR$ in a circle $x^2+y^2=16$ such that $Q\equiv (2\sqrt{2},2\sqrt{2})$ and $R\equiv (-2,2\sqrt{3}).$ Then the measure of $\angle QPR$ is

• A. ${75}^{\circ }$
• B. ${45}^{\circ }$
• C. ${37.5}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is C ${37.5}^{\circ }$


Slope of $OQ:$ $m=1$ (makes an angle ${45}^{\circ }$ with $+$ve $x$-axis)
Slope of $OR:$ $m^{\prime }=-\sqrt{3}$ (makes an angle ${120}^{\circ }$ with $+$ve $x$-axis as $R$ lies in $2$nd quadrant.)
$\therefore \angle QOR={120}^{\circ }-{45}^{\circ }={75}^{\circ }$
$\therefore \angle QPR={37.5}^{\circ }$ ($\because$ angle subtended by an arc at the centre of a circle is twice the angle subtended at its circumference)