Question

i) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the x-axis.

ii) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the y-axis

Answer 1

i) Tangent line:

$(y-y_0)=y'(x_0)(x-x_0)$

$y'=x+y^2+x(1+2yy')$

$y'=\frac{x+y^2+x}{1-2xy}$

Line is parallel to the x-axis:

$y=c=const\implies y'=0$

$y'=\frac{x+y^2+x}{1-2xy}=0$

Tangent line is parallel to the x-axis if :

$2x+y^2=0$

$2xy\neq1$

ii) Tangent line is parallel to the y-axis if :

$y'\to \infin$

So we have:

$2xy=1$