Question

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

• A. 2
• B. 4
• C. 0
• D. 3

Answer 1

Answer: (B)

4

Answer 2

Normal of line is parallel to line x+90y+2=0
$m_N​=−\frac{1}{90}​$
$−(\frac{dx}{dy}​)_{(x_1​y_1​)​}=−\frac{1}{90}⇒(\frac{dy}{dx}​)_{(x_1​y_1​)​}=90$
Now,
$\frac{dy}{dx}​=270x^4−540x^3−210x^2+360x+210=90$
$⇒x=1,2,\frac{−2}{3}​,\frac{−1}{3}​$
so, the correct option is(B): 4 normals