### Application Of Derivatives Question 6

#### Question 6 - 30 January - Shift 1

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 :

(1) 2

(2) 3

(3) 4

(4) 0

## Show Answer

#### Answer: (3)

#### Solution:

#### Formula: Normal to a curve

Normal of line is parallel to line $x+90 y+2=0$

$m_N=-\frac{1}{90}$

$-(\frac{d x}{d y}) _{(x_1 y_1)}=-\frac{1}{90} \Rightarrow(\frac{d y}{d x}) _{(x_1 y_1)}=90$

Now,

$\frac{d y}{d x}=270 x^{4}-540 x^{3}-210 x^{2}+360 x+210=90$

$\Rightarrow x=1,2, \frac{-2}{3}, \frac{-1}{3}$

Total number of points is 4