Question

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

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

Answer 1

Answer: (C)

$\frac{\pi }{4}$

Equation of the circle is $x^2+y^2=25;$

$△PQR$ is inscribed in the circle and co-ordinates of $Q$ and $R$ are $(3,4)$ and $(-4,3).$

We know that the slope of $OQ\left(m_1\right)=\frac{4}{3}$ and the slope of $OR\left(m_2\right)=\frac{-3}{4}$

Since $m_1\times m_2=\frac{4}{3}\times (-\frac{3}{4})=-1,$

therefore $\angle QOR=\frac{\pi }{2}.$

We also know that the angle subtended at the centre of a circle is twice the angle subtended in the alternate seg-ment.

Therefore,$\angle QPR=\frac{\pi }{4}$