Question

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

Answer 1

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, $m=\pm2$