The area of the triangle formed by the positive $x$-axis and the normal and the tangent to the circl

Question

The area of the triangle formed by the positive $x$-axis and the normal and the tangent to the circle $x^{2}+y^{2}=4$ at $(1, \sqrt{3})$ is

Professor PC Thomas Classes · Answer

Centre of the circle is the origin, radius $O P=2$\ Equation of the tangent at $P(1, \sqrt{3})$ is $S_{1}=0$ i.e., $x+y \sqrt{3}=4$\ This cuts the $x$-axis at $Q(4,0)$\ $	herefore \quad$ Area of $	riangle O P Q=\frac{1}{2} 	imes O P 	imes P Q$\ $(\because O P \perp P Q)$\ $=\frac{1}{2} 	imes 2 	imes 2 \sqrt{3}=2 \sqrt{3}$ sq. units.\