Question

Let $M$ and $N$ be the number of points on the curve $y^5-9 x y+2 x=0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M + N$ equals _____

Answer 1

$y^5-9 xy +2 x =0 $
$ 5 y^4 \frac{d y}{x}-9 x \frac{d y}{d x}-9 y+2=0$
$ \frac{d y}{d x}\left(5 y^4-9 x\right)=9 y-2$
$ \frac{d y}{d x}=\frac{9 y-2}{5 y^4-9 x}=0 $ (for horizontal tangent) $y=\frac{2}{9} \Rightarrow $ Which does not satisfy the original equation $\Rightarrow M =0$
Now $5 y ^4-9 x =0 $ (for vertical tangent)
$ 5 y ^4(9 y -2)-9 y ^5=0$
$ y ^4[45 y -10-9 y ]=0 $
$ y =0 (Or) 36 y =10 $
$ y=\frac{5}{18}$
$ y=0 \Rightarrow x=0 \& y=\frac{5}{18} \Rightarrow x= $
$(0,0) \left(x, \frac{5}{18}\right)$
$N =2$
$ M + N =0+2=2$