Points on curve where tangent are equally inclined

Question

What are the points on the curve $ x^{3/2} + y^{3/2} = a^{3/2} $ where the tangents are equally inclined to the axes?

Answer 1

$$x^{3/2}+y^{3/2}=\frac{a^3}{2}\Longrightarrow \frac{3}{2}\left(x^{1/2}\,dx+ y^{1/2}\,dy\right)=0\Longrightarrow$$

$$y'=-\sqrt\frac{x}{y}\Longrightarrow \,\text{two points}(x_1,y_1)\,,\,(x_2,y_2)\,\,\text{fulfill the condition}\Longleftrightarrow \frac{x_1}{x_2}=\frac{y_1}{y_2}$$

-- I'll leave it to you to check what happens when $\,x=0\,\,\vee\,\,\,y=0\,$ --

Since it must be that $\,x,y\geq 0\,$ (why?) , and also

$$y=\sqrt[3]{\left(\frac{a^3}{2}-x^{3/2}\right)^2}$$ we get

$$\frac{x_1}{x_2}=\frac{y_1}{y_2}=\left(\frac{\frac{a^3}{2}-x_1^{3/2}}{\frac{a^3}{2}-x_2^{3/2}}\right)^{2/3}\stackrel{\text{after a little algebra}}\Longleftrightarrow x_1^{3/2}=x_2^{3/2}$$

so...