Normal to a curve at the point x=1

Question

Find the equation to the normal of the curve $y=x-\frac{2}{x^2}$ at the point where $x=1$. Then show that the normal does not meet the curve again.

I'm not sure what to do. I understand that you differentiate then sub in the x value but as there is no Y value im not sure how to do it. Any ideas?

Answer 1

HINT

We have

• $y=x-\frac{2}{x^2}\implies y(1)=-1$
• $y'=1+\frac 4 {x^3}\implies y'(1)=5$

then the normal at $P(1,-1)$ is

$$y-(-1)=-\frac15(x-1)\implies y=-\frac15 x-\frac45$$

Now we need to show that the interscetion

• $y=x-\frac{2}{x^2}$
• $y=-\frac15 x-\frac45$

has only one solution.