Question

The absolute value of slope of common tangents to parabola $y^2=8x$ and hyperbola $3x^2-y^2=3$ is

Answer 1

Answer: 2

Here we need to find the common tangent between the two curves. So, write standard equation of tangent of slope $m$ for both the curves and then compare it to find the value of $m$ .
The equation of tangent to given parabola $y^2=8x$ is $y=mx+\frac{2}{m}$ .
(The equation of tangent to parabola $y^2=4ax$ is $y=mx+\frac{a}{m}$ )
And the equation of tangent to the given hyperbola $\frac{x^2}{1}-\frac{y^2}{3}=1$ is $y=mx\pm \sqrt{m^2-3}$ .
(The equation of tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $y=mx\pm \sqrt{m^2{a}^2-b^2}$ )
Now, on comparing the above equations of tangents to both the given curves, we get
$\Rightarrow m^2-3=\frac{4}{m^2}$
$\Rightarrow m^4-3m^2-4=0$
$\Rightarrow m^2=4,-1$
Therefore, $m=\pm 2$