A common tangent to the conics x^2=6y and 2x^2-4y^2=9 is:

Question

A common tangent to the conics $x^2=6y$ and $2x^2-4y^2=9$ is:

Accepted Answer

Try the line $x-y=	frac32$, i.e. $y=x-	frac32$.

In $x^2=6y$: $x^2=6x-9\Rightarrow x^2-6x+9=0\Rightarrow(x-3)^2=0$ --- tangent at $x=3,\ y=	frac32$.

In $2x^2-4y^2=9$: $2x^2-4\left(x-	frac32ight)^2=9\Rightarrow 2x^2-4x^2+12x-9=9\Rightarrow-2x^2+12x-18=0\Rightarrow x^2-6x+9=0\Rightarrow(x-3)^2=0$ --- also tangent at the same point.

Hence $x-y=	frac32$ is a common tangent.

Solutions · 1