Let $C$ be the curve $y^{3}-3 x y+2=0$. If $H$ is the set of points on the curve $C$ when the tangen

Question

Let $C$ be the curve $y^{3}-3 x y+2=0$. If $H$ is the set of points on the curve $C$ when the tangent is horizontal and $v$ be the set of all points on the curve $C$ when the tangent is vertical then $H=$ and $v=$

Professor PC Thomas Classes · Answer

$y^{3}-3 x y+2=0$ Differentiating wrt $x$, we get\ $\frac{d y}{d x}=\frac{y_{1}}{y_{1}^{2}-x_{1}}$\ At point $P\left(x_{1}, y_{1}ight)$ on the curve slope of horizontal line is zero.\ $\Rightarrow \frac{y_{1}}{y_{1}^{2}-x_{1}}=0 \Rightarrow y_{1}=0$ There are no points on the curve whose ordinate $=0$\ $	herefore \quad H=\phi$\ Slope of vertical line $=\infty$\ $\Rightarrow \quad \frac{y_{1}{ }^{2}-x_{1}}{y_{1}}=0 \Rightarrow x_{1}=y_{1}^{2}$\ Solving with $y_{1}{ }^{3}-3 x_{1} y_{1}+2=0$\ the points is $(1,1)$\ $	herefore \quad H=\phi$ and $v=\{(1,1)\}$.\