曲线 $C$ 的极坐标方程为 $r=a\tan\big(\tfrac18\theta\big)$，其中 $a$ 为正常数，且 $0\leqslant\theta\leqslant2\pi$。

(a) 画出 $C$ 的草图，并写出 $C$ 上的点到极点的最大距离，以 $a$ 表示。

(b) 求由 $C$ 与极轴围成的区域的面积，以 $a$ 表示。

(c) 证明：在 $C$ 上距离极轴最远的点处，

并验证该方程在 $4.95$ 与 $5$ 之间有一个根。

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。

极径从0单调增加到a，角度逆时针转一整圈，得到从极点向外展开的螺旋形曲线。端点与极轴的切线方向也要正确。

For $0\leqslant\theta\leqslant2\pi$,

Also, $r(0)=0$ and $r(2\pi)=a$. Sketch an anticlockwise outward spiral starting at the pole and ending at $(a,0)$ after one full turn. It crosses the negative $x$-axis once, at $(-a\tan(\pi/8),0)$, and has no self-intersection away from the pole.

At the pole the tangent is the initial line. At $\theta=2\pi$,

so the final tangent has gradient $4$; the curve approaches $(a,0)$ from below.

Since $r$ increases throughout, the greatest distance from the pole is $\boxed{a}$.

曲线转过一整圈，区域由整段极坐标曲线和正极轴线段封闭。将正切平方化成正割平方减1后积分。

The boundary runs from the pole along $C$ to $(a,0)$ and back along the initial line. Its area is

到极轴所在直线的垂距是纵坐标绝对值。最大值不在纵坐标为零的端点，因此该点纵坐标的导数为零。整理时乘上恒正的因子，再用连续函数变号验证根区间。

The perpendicular distance from the initial line is $|y|$, where

At $\theta=0,2\pi$ this distance is zero, while some interior points have positive distance. Hence the maximum is attained at an interior point where $y\ne0$ and $\mathrm{d}y/\mathrm{d}\theta=0$.

Differentiating gives

Multiplying the stationary equation by $8\cos^2(\theta/8)/a>0$ gives

Using $2\sin u\cos u=\sin2u$, this becomes

Let $F(\theta)=4\sin(\theta/4)\cos\theta+\sin\theta$. In radians,

Since $F$ is continuous, it has a root in $\boxed{(4.95,5)}$.