"The number of points on the curve $y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$ at which the normal lines are parallel to $x+90 y+2=0$ is :"

Question

Accepted Answer

"

$y'=270x^4-540x^3-210x^2+360x+210$

Slope of normal $=-\frac{1}{90}$

$\therefore$ Slope of tangent = 90

$\therefore$ Number of normal will be number of solutions of

$270x^4-540x^3-210x^2+360x+210=90$

$\Rightarrow 9x^4-18x^3-7x^2+12x+4=0$

$\therefore x=1,2,-\frac{1}{3},-\frac{2}{3}$ are roots

"

The number of points on the curve $y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$ at which the normal lines are parallel to $x+90 y+2=0$ is :