Question

Find the coordinates of the points on the curve $y=x-\frac{4}{x}$, where the tangents are parallel to the line $y=2x$.

Answer 1

Equation of line is given by,
$y=2x$ .......... (1)
We know that
$y=mx+c$ .......... (2)
On comparing equation (2) with (1),
We get, $m=2$
Given, $y=x-\frac{4}{x}$
Differentiating w.r.t.$x$, we get
$\frac{dy}{dx}=1-4(-1).\frac{1}{x^2}$
$=1+\frac{4}{x^2}=m$ ......... (3)
Slope of the tangent, $m=1+\frac{4}{x^2}$
$\Rightarrow 1+\frac{4}{x^2}=2$
$\Rightarrow x^2+4=2x^2$
$\Rightarrow x=\pm 2$ and $y=2x$
$\Rightarrow x_1=2,x_2=-2$
$\Rightarrow y_1=4,y_2=-4$
Co-ordinates of the point of contact are $(2,4)$ and $(-2,-4)$.