Question

Find the points on the curve $xy+4=0$ at which the tangents are inclined at an angle of ${45}^o$ with the $x$-axis.

Answer 1

Given the curve is $xy+4=0$

or, $y=-\frac{4}{x}$.

Now $\frac{dy}{dx}=\frac{4}{x^2}$.

We've the slope of the tangent at $(x,y)$ is $\frac{dy}{dx}$.

According to the problem,

$\frac{dy}{dx}=\tan {45}^o$

or, $\frac{4}{x^2}=1$

or, $x^2=4$

or, $x=\pm 2$.

If $x=\pm 2$ then $y$ will be $\mp 2$.

So the points are $(\pm 2,\mp 2)$.