ON THE OUTLINES OF PLANE CURVES OF THE FORM $(ax)^{\alpha}+(by)^{\alpha}=r^{\alpha}$ WITH $\alpha>0$

We consider plane curves of the form $(ax)^{\alpha}+(by)^{\alpha}=r^{\alpha}$ defined on the first quadrant of $\mathbb R^2$, where $\alpha>0$ and $a,b,r>0$. We summarize the outlines of them by using elementary differential calculus. We will in this note understand that they are classified into three types of curves, convex, straight and concave, depending on $\alpha$.