Question:
The normal at the point $(1,1)$ on the curve $2 y+x^{2}=3$ is
(A) $x+y=0$
(B) $x-y=0$
(C) $x+y+1=0$
(D) $x-y=1$
Solution:
The equation of the given curve is $2 y+x^{2}=3$.
Differentiating with respect to x, we have:
$\frac{2 d y}{d x}+2 x=0$
$\Rightarrow \frac{d y}{d x}=-x$
$\left.\therefore \frac{d y}{d x}\right]_{(1,1)}=-1$
The slope of the normal to the given curve at point (1, 1) is
$\frac{-1}{\left.\frac{d y}{d x}\right]_{(1,1)}}=1$
Hence, the equation of the normal to the given curve at (1, 1) is given as:
$\Rightarrow y-1=1(x-1)$
$\Rightarrow y-1=x-1$
$\Rightarrow x-y=0$
The correct answer is B.