The normal at the point

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.