Question

The equation of normal to the curve $x^2+y^2=r^2$ at $p\left(\theta \right)$ is

• A. $x\sin \theta -y\cos \theta =0$
• B. $x\sin \theta +y\cos \theta =0$
• C. $x\cos \theta -y\sin \theta =0$
• D. $x\cos \theta +y\sin \theta =0$

Answer 1

The correct option is A $x\sin \theta -y\cos \theta =0$

Curve :- $X^2+y^2=r^2$

$2x+2y\frac{dy}{dx}=0$

$\Rightarrow \frac{dy}{dx}=-\frac{x}{y}$

$=\frac{-r\cos \theta }{r\sin \theta }$

$=\frac{-\cos \theta }{\sin \theta }$

$\Rightarrow -\frac{dx}{dy}=\frac{\sin \theta }{\cos \theta }$

$\therefore$ Equation of normal to the curve is

$(y-r\sin \theta )$=$\frac{\sin \theta }{\cos \theta }(x-r\cos \theta )$

$\Rightarrow y\cos \theta -r\sin \theta \cos \theta =x\sin \theta -r\cos \theta \sin \theta$

$\Rightarrow x\sin \theta -y\cos \theta =0$