Question

The equation of normal to the curve $x+y=x^y$, where it cuts x-axis is

• A. $y=x+1$
• B. $y=-x+1$
• C. $y=x-1$
• D. $y=-x-1$

Answer 1

The correct option is C $y=x-1$

$ln(x+y)=yln\left(x\right)$

$\frac{1}{x+y}.(1+\frac{dy}{dx})=\frac{y}{x}+\frac{dy}{dx}.ln\left(x\right)$

Now $y=0$ and $\frac{dy}{dx}=m$ we get

$\frac{1+m}{x}=m.ln\left(x\right)$

Now $x=1$ when $y=0$

Hence

$1+m=0$

$m=-1$

Slope of normal $=1.$

Hence

$\frac{y-0}{x-1}=1$

$y=x-1$.