Question

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

• A. 2
• B. 4
• C. 0
• D. 3

Answer 1

Answer: (B)

Normal of line is parallel to line $x+90y+2=0$
$m_N=-\frac{1}{90}$
$-{\left(\frac{dx}{dy}\right)}_{\left(x_1{y}_1\right)}=-\frac{1}{90}\Rightarrow {\left(\frac{dy}{dx}\right)}_{\left(x_1{y}_1\right)}=90$
Now,
$\frac{dy}{dx}=270x^4-540x^3-210x^2+360x+210=90$
$\Rightarrow x=1,2,\frac{-2}{3},\frac{-1}{3}$
(4) normals