Question

The $△PQR$ is inscribed in the circle $x^2+y^2=25$. If $Q$ and $R$ have coordinates $(3,4)$ and $(-4,3)$, respectively. Then, $\angle QPR$ is equal to

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is A $\frac{\pi }{2}$

Let $O$ is the point on centre and $P$ is the point on circumference.

Therefore, angle $QOR$ is double the angle $QPR$. So, it is sufficient to find the angle $QOR$
Now, slope of $OQ,m_1=\frac{4-0}{3-0}=\frac{4}{3}$
Slope of $OR,m_2=\frac{3-0}{-4-0}=-\frac{3}{4}$
Again, $m_1{m}_2=-1$
Therefore, $\angle QOR={90}^{\circ }$
Which implies that $\angle QPR=\frac{{90}^{\circ }}{2}={45}^{\circ }$.