一条曲线的方程为 $2y^3-3x^2y-x^3=16$。

(a) 证明 $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x^2+2xy}{2y^2-x^2}$。

(b) 由此求曲线上法线平行于 $y$ 轴的点的坐标。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

隐式求导时，$x^2y$ 要用乘积法则。把所有含导数的项收集在一起再约去公共系数。

Differentiating implicitly,

Collecting derivative terms,

Thus

法线竖直意味着切线水平，因此导数的分子为零。求得点后，检查分母非零并代回曲线。

A normal parallel to the $y$-axis corresponds to a horizontal tangent. Hence, by (a),

If $x=0$, the curve gives $2y^3=16$, so $y=2$.

If $x=-2y$, substitution gives

so $-2y^3=16$, giving $y=-2$ and $x=4$.

The derivative denominators at these points are respectively $8$ and $-8$, both nonzero. Therefore the required points are