Question

Consider the parabola $y^2=8x$ and the circle $x^2+y^2=2$.

Which of the following is true for these curves?

• A. They have only two common tangent which are mutually perpendicular.
• B. They have only two common tangents which are parallel to each other.
• C. They have infinitely many common tangents.
• D. They does not have any common tangents.

Answer 1

Answer: (A)

They have only two common tangent which are mutually perpendicular.

If common tangent is $y=mx+c,$ then

$c=\frac{2}{m}$ and $c^2=2(1+m^2)$

If $m^2=t,$ then

$\frac{4}{t}=2(1+t)\Rightarrow 2t^2+2t-4=0$

or $(t+2)(t-1)=0$

$\therefore t=1(t\ne -2)$

Thus, $m=\pm 1$

Hence, tangents are $y=x+c$ and $y=-x+c$ which are perpendicular to each other.

Hence, option A.