Question

The equation of common tangent to $y^2=4ax$ and $x^2=4ay$ is.

• A. $x+y+a=0$
• B. $x-y+a=0$
• C. $x-y-a=0$
• D. $x+y-a=0$

Answer 1

Answer: (A)

$x+y+a=0$

The equation of any tangent to $y^2=4ax$ is

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

If it touches $x^2=4ay$ then, the equation.

$x^2=4a(mx+\frac{a}{m})$ has equal roots.

$m{x}^2-4a{m}^{2}x-4a^2=0$ has equal roots

If the roots are equal, the discriminant is equal to $0$

i.e.,$b^2-4ac=0$

$16a^2{m}^4+16a^{2}m=0$

$16a^2{m}^4=-16a^{2}m$

$m=-1$

Putting $m=-1$ in $y=mx+am$ we get,

$y=-x-a$

$x+y+a=0$ This is the common tangent to the two parabola.