Question

$2$ common tangents to the circle $x^2+y^2=2a^2$ and equation of parabola $y^2=8ax$ are

• A. $x=\pm (y+2a)$
• B. $y=\pm (y+2a)$
• C. $x=\pm (y+a)$
• D. $y=\pm (x+a)$

Answer 1

The correct option is D $y=\pm (x+a)$

Consider the given equation

Equation of circle $x^2+y^2=2a^2......\left(1\right)$

Equation of parabola is $y^2=8ax......\left(2\right)$

We know that equation of tangent

$y=mx+c$

Where $c=\sqrt{2a(1+m^2)}$ (For given Circle equation)

Where $c=\frac{2a}{m}$ (For given parabola equation)

Then,

Equation of Circle is $y=mx+\sqrt{2a}\sqrt{1+m^2}......\left(3\right)$

Equation of Parabola is $y=mx+\frac{2a}{m}......\left(4\right)$

By equation (3) and (4) to and we get,

$\frac{2a}{m}=\sqrt{2}a\sqrt{1+m^2}$

$\frac{4a^2}{m^2}=2a^2(1+m^2)$

$2=m^4+m^2$

$m^4+m^2-2=0$

$m^4+(2-1)m^2-2=0$

$m^4+2m^2-1m^2-2=0$

$m^2(m^2+2)-1(m^2+2)=0$

$(m^2+2)(m^2-1)=0$

If

$m^2-1=0$

$m^2=1$

$m=\pm 1$

Put the value of $m$ in equation (1) and (2) to we get,

Equation of common tangents:

$y=-x-2a$

$y=x+2a$

OR

$y=\pm (x+2a)$

Hence, this is the answer.