Question

If the line $ax+y=c,$ touches both the curves $x^2+y^2=1$ and $y^2=4\sqrt{2}x,$ then $|c|$ is equal to :

• A. $\sqrt{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

The correct option is A $\sqrt{2}$

The equation of tangent to the parabola $y^2=4ax$ with slope $m$ is
$y=mx+\frac{a}{m}$
The line $ax+by+c=0$ touches the circle $x^2+y^2=r^2,$ then
$\frac{|c|}{\sqrt{a^2+b^2}}=r$

Now, tangent to the curve $y^2=4\sqrt{2x}$ is
$y=mx+\frac{\sqrt{2}}{m}$

This line is also tangent to the circle $x^2+y^2=1$
$\therefore \left|\frac{\sqrt{2}/m}{\sqrt{1+m^2}}\right|=1\Rightarrow m^4+m^2-2=0\Rightarrow (m^2+2)(m^2-1)=0\Rightarrow m=\pm 1$

So, the equation of tangents are
$y=x+\sqrt{2}$ and $y=-x-\sqrt{2}$

Compare with $y=-ax+c$
$\Rightarrow a=\mp 1\text{and}c=\pm \sqrt{2}\therefore |c|=\sqrt{2}$