Conic Sections圆锥曲线

• Distance.距离公式。 Direct application of the Pythagorean theorem to the right triangle whose legs are the differences in coordinates.把坐标差作直角边，对相应直角三角形应用勾股定理即得。
• Midpoint.中点公式。 Coordinate averages. Useful for finding centres of circles when two endpoints of a diameter are known.坐标分量平均。已知直径两端点时用以求圆心。
• Circle.圆方程。 $(x-h)^{2} + (y-k)^{2} = r^{2}$ , the locus of points at distance $r$ from centre $(h, k)$. This is CCSSM HSG-GPE.A.1 verbatim , the standard explicitly names "deriving the circle equation via the Pythagorean theorem" as the expected approach.$(x-h)^{2} + (y-k)^{2} = r^{2}$ , 到圆心 $(h, k)$ 距离为 $r$ 的点的轨迹。这正是 CCSSM HSG-GPE.A.1，标准明确以"由勾股定理推导圆方程"作为期望路径。
• General form.一般形式。 $x^{2} + y^{2} + Dx + Ey + F = 0$ is a circle iff the implied radius $r = \tfrac{1}{2}\sqrt{D^{2} + E^{2} - 4F}$ is real and positive. Recover centre / radius by completing the square in $x$ and $y$ separately.$x^{2} + y^{2} + Dx + Ey + F = 0$ 为圆当且仅当隐含半径 $r = \tfrac{1}{2}\sqrt{D^{2} + E^{2} - 4F}$ 为正实数。分别对 $x$、$y$ 配方即可读出圆心 / 半径。

A parabola is the locus of points equidistant from a fixed point (the focus, $F$) and a fixed line (the directrix). CCSSM HSG-GPE.A.2 names this definition explicitly and asks students to derive the equation from it.抛物线是与一固定点（焦点 $F$）和一固定直线（准线 directrix）等距的点的轨迹。CCSSM HSG-GPE.A.2 正是以此定义为基础，要求学生由它推导方程。

• Vertical-axis parts.竖轴抛物线各要素。 Vertex $(h, k)$, focus $(h, k + p)$, directrix $y = k - p$. Opens up if $p > 0$, down if $p < 0$.顶点 $(h, k)$，焦点 $(h, k + p)$，准线 $y = k - p$。$p > 0$ 开口向上，$p < 0$ 开口向下。
• Horizontal-axis parts.横轴抛物线各要素。 Vertex $(h, k)$, focus $(h + p, k)$, directrix $x = h - p$. Opens right if $p > 0$, left if $p < 0$.顶点 $(h, k)$，焦点 $(h + p, k)$，准线 $x = h - p$。$p > 0$ 开口向右，$p < 0$ 开口向左。
• Vertex form.顶点式。 $y = a(x - h)^{2} + k$ with $a = \frac{1}{4p}$ is the same equation written in function-of-$x$ form , this is the version Ontario MCR3U / BC PC 11 / AB Math 20-1 use.$y = a(x - h)^{2} + k$（其中 $a = \frac{1}{4p}$）是同一方程以 $x$ 的函数形式书写 , 安大略 MCR3U / BC PC 11 / 阿尔伯塔 Math 20-1 都采用此版本。

An ellipse is the locus of points $P$ such that $|PF_{1}| + |PF_{2}| = 2a$, where $F_{1}$ and $F_{2}$ are the two foci. The standard form (centred at origin, major axis horizontal) is:椭圆是满足 $|PF_{1}| + |PF_{2}| = 2a$ 的点 $P$ 的轨迹，$F_{1}$、$F_{2}$ 为两焦点。以原点为中心、主轴水平的标准方程为：

• Vertices.主顶点。 $(\pm a, 0)$ on the major axis.主轴方向上的 $(\pm a, 0)$。
• Co-vertices.短轴端点。 $(0, \pm b)$ on the minor axis.短轴方向上的 $(0, \pm b)$。
• Foci.焦点。 $(\pm c, 0)$ on the major axis, where $c = \sqrt{a^{2} - b^{2}}$.主轴方向上的 $(\pm c, 0)$，其中 $c = \sqrt{a^{2} - b^{2}}$。
• Eccentricity.离心率。 $$e = \tfrac{c}{a}, \quad 0 \le e < 1.$$ $e = 0$ is a circle; $e \to 1$ is a very elongated ellipse.$e = 0$ 即圆；$e \to 1$ 椭圆极扁长。
• Vertical major axis.主轴竖直情形。 If $b > a$, swap roles: $\tfrac{x^{2}}{b^{2}} + \tfrac{y^{2}}{a^{2}} = 1$ with $a^{2}$ now under $y^{2}$ and foci on the $y$-axis. (Some texts always reserve $a$ for the longer semi-axis , we follow that convention.)若 $b > a$，互换：$\tfrac{x^{2}}{b^{2}} + \tfrac{y^{2}}{a^{2}} = 1$，$a^{2}$ 现在在 $y^{2}$ 下方，焦点在 $y$ 轴。（部分教材始终用 $a$ 表示更长半轴 , 本指南采用该约定。）

A hyperbola is the locus of points $P$ such that $\big| |PF_{1}| - |PF_{2}| \big| = 2a$ , the absolute difference of distances to two foci is constant. The standard form (centred at origin, transverse axis horizontal) is:双曲线是满足 $\big| |PF_{1}| - |PF_{2}| \big| = 2a$ 的点 $P$ 的轨迹 , 到两焦点距离之差（绝对值）为常数。以原点为中心、实轴水平的标准方程为：

• Vertices.顶点。 $(\pm a, 0)$ on the transverse axis. The two branches open outward through these vertices.实轴上的 $(\pm a, 0)$。两支由这两顶点向外延展。
• Foci.焦点。 $(\pm c, 0)$ with $c = \sqrt{a^{2} + b^{2}}$ , note the plus sign, opposite to the ellipse.$(\pm c, 0)$，$c = \sqrt{a^{2} + b^{2}}$ , 注意是加号，与椭圆相反。
• Asymptotes.渐近线。 $$y = \pm \tfrac{b}{a} x.$$ The branches approach but never touch these lines as $|x| \to \infty$.$|x| \to \infty$ 时两支无限趋近但不触及。
• Eccentricity.离心率。 $e = c/a > 1.$ Always greater than $1$ , that's what separates hyperbolas from ellipses on the eccentricity scale.总大于 $1$ , 这正是离心率刻度上双曲线与椭圆的分界。
• Vertical transverse axis.实轴竖直情形。 $$\tfrac{y^{2}}{a^{2}} - \tfrac{x^{2}}{b^{2}} = 1$$ with vertices on the $y$-axis at $(0, \pm a)$ and asymptotes $y = \pm \tfrac{a}{b} x$.顶点 $(0, \pm a)$，渐近线 $y = \pm \tfrac{a}{b} x$。

When this represents a non-degenerate conic (i.e. an actual curve, not a single point / pair of lines / empty set), compute the discriminant $\Delta = B^{2} - 4AC$:当此方程表示非退化的圆锥曲线（即真实曲线，而非单点 / 两直线 / 空集）时，计算判别式 $\Delta = B^{2} - 4AC$：

• Without the $xy$ term ($B = 0$).无 $xy$ 项（$B = 0$）。 $\Delta = -4AC$. So: $AC > 0$ ellipse / circle; $AC = 0$ (one of $A, C$ is zero) parabola; $AC < 0$ hyperbola.$\Delta = -4AC$。所以 $AC > 0$ 椭圆 / 圆；$AC = 0$（$A$、$C$ 之一为 $0$）抛物线；$AC < 0$ 双曲线。
• When $B \ne 0$.$B \ne 0$ 时。 The conic is rotated. The discriminant still classifies the type; rotating the axes by an angle $\theta$ with $\cot(2\theta) = (A - C)/B$ eliminates the $xy$ term.曲线旋转了。判别式仍能定型；以满足 $\cot(2\theta) = (A - C)/B$ 的角 $\theta$ 旋转坐标轴可消去 $xy$ 项。

Replacing $x$ with $(x - h)$ shifts the curve $h$ units right (so the centre moves from $(0, 0)$ to $(h, 0)$). Replacing $y$ with $(y - k)$ shifts it $k$ units up. The same rule applies to every conic, exactly as it did in Unit 10 for general function transformations.把 $x$ 换为 $(x - h)$ 使曲线向右平移 $h$ 单位（中心由 $(0, 0)$ 移至 $(h, 0)$）。把 $y$ 换为 $(y - k)$ 使其向上平移 $k$ 单位。所有圆锥曲线遵循同一规则，与 Unit 10 中函数变换的处理完全一致。