Question

The set of values of $m$ for which a chord of slope $m$ of the circle $x^2+y^2=16$ touches the parabola, $y^2=8x$

• A. $(-\infty ,-1)\cup (1,\infty )$
• B. $(-\infty ,\infty )$
• C. $(-\infty ,-\sqrt{\frac{\sqrt{2}-1}{2}})\cup (\sqrt{\frac{\sqrt{2}-1}{2}},\infty )$
• D. $(-1,1)$

Answer 1

Answer: (C)

$(-\infty ,-\sqrt{\frac{\sqrt{2}-1}{2}})\cup (\sqrt{\frac{\sqrt{2}-1}{2}},\infty )$

Equation of tangent to parabola $y^2=4ax$ is $y=mx+\frac{a}{m}$

Then, the equation of tangent to parabola $y^2=8x$ is

$y=mx+\frac{2}{m}$

For chord, perpendicular distance has to be less than radius

$\Rightarrow \frac{\left|\frac{2}{m}\right|}{\sqrt{1+m^2}}<4$

$\Rightarrow \frac{1}{m^2}<4(1+m^2)$

$\Rightarrow 4m^4+4m^2-1>0$

$\Rightarrow mϵ(-\infty ,-\sqrt{\frac{\sqrt{2}-1}{2}})\cup (\sqrt{\frac{\sqrt{2}-1}{2}},\infty )$