Question

The equation of the normals to the curve $y=2x^3+2x$ which are parallel to $2x+16y=7$ is

• A. $x+8y=33,x+8y=-33$
• B. $x+8y=31,x+8y=-31$
• C. $2x+16y=33,2x+16y=-33$
• D. $2x+16y=31,2x+16y=-31$

Answer 1

The correct option is A $x+8y=33,x+8y=-33$

The curve is $y=2x^3+2x$
$\frac{dy}{dx}=6x^2+2$
The normal is parallel to the line $2x+16y=7$
So $\frac{-1}{dy/dx}=\frac{-2}{16}$
$\Rightarrow \frac{-1}{6x^2+2}=\frac{-1}{8}$
$\Rightarrow 6x^2+2=8$
$\Rightarrow x^2=1\Rightarrow x=\pm 1$
$\Rightarrow y=\pm 4$
So the point of contact are
$(1,4)$ and $(-1,-4)$

Hence, required normals are
$y-4=\frac{-1}{8}(x-1)$
$\Rightarrow 8y-32=-x+1$
$\Rightarrow x+8y=33$

$y+4=\frac{-1}{8}(x+1)$
$\Rightarrow 8y+32=-x-1$
$\Rightarrow x+8y=-33$