UPSC NDA II 2024 Mathematics Paper

Concept:Angle subtended by a chord at the centre is double the angle subtended by the same chord at any point on the remaining part of the circle.Explanation:The given circle is $x^2+y^2=100$, so radius $OB=OC=10$ units.Length of chord $BC$ is $10\sqrt{2}$ (derived from coordinates).In $\triangle BOC$, $OB^2+OC^2=100+100=200$ and $BC^2=(10\sqrt{2})^2=200$.Thus $OB^2+OC^2=BC^2$, so by Pythagoras theorem, $\angle BOC=90^\circ$.Since angle subtended by chord $BC$ at the centre ($\angle BOC$) is twice the angle at any point $A$ on the remaining circle, we have $\angle BAC = \frac{1}{2}\angle BOC = 45^\circ = \frac{\pi}{4}$.However, for a point $A$ on the other arc (the major arc), the angle is supplementary: $\angle BAC = 180^\circ-45^\circ = 135^\circ = \frac{3\pi}{4}$.Answer:$\frac{\pi}{4}$ or $\frac{3\pi}{4}$ (Option C).