Implicit differentiation

Question

The equation of a curve is $xy(x + y) = 2a^3$ , where $a$ is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the x-axis, and find the coordinates of this point.

I am completely lost.

Answer 1

The curve is $x^2y+xy^2=2a^3$. Differentiating, we get $2xy+x^2y'+y^2+2xyy'=0$ using product rule.
Now set $y'$ equal to $0$. You will get $y=0$ or $y+2x=0$.
$y=0$ is not possible because putting $y=0$ in original equation contradicts the given fact that $a$ is nonzero. Now put $-2x=y$ in original equation. You will get the point as $(a,0)$.