Conic Section

The One Thing

Conic sections are loci defined by a focus–directrix distance ratio, $PF/PM=e$, and classified by eccentricity: a parabola has $e=1$, an ellipse has $0<e<1$, and a hyperbola has $e>1$. Their standard equations determine orientation, symmetry, vertices, foci, directrices, axes, latus recta, and, for hyperbolas, asymptotes.

Definitions and Results

• Conic Section: The locus generated by the intersection of a plane with a double cone; it may form a circle, ellipse, parabola, or hyperbola.
• Focus: A fixed point used in defining a conic. A parabola has one focus, whereas an ellipse and hyperbola have two.
• Directrix: A fixed straight line used with a focus to define a conic.
• Eccentricity: The constant ratio of the distance of a point on the conic from the focus to its perpendicular distance from the directrix:

$$\frac{PF}{PM}=e.$$ Here, $P$ is a point on the conic, $F$ is the focus, and $M$ is the foot of the perpendicular from $P$ to the directrix.

• Parabola: The locus of a point whose distance from a fixed focus equals its distance from a fixed directrix. Thus, $e=1$.
• Ellipse: The locus of a point for which the sum of its distances from two fixed points, called foci, is constant. Its eccentricity satisfies $0<e<1$.
• Hyperbola: The locus of a point for which the absolute difference of its distances from two fixed points, called foci, is constant. Its eccentricity satisfies $e>1$.
• Axis: A line of symmetry of a conic. An ellipse has major and minor axes; a hyperbola has transverse and conjugate axes.
• Vertex: A point where the conic meets its principal axis. A parabola has one vertex, while an ellipse and hyperbola have vertices on their principal axes.
• Latus Rectum: A chord through a focus and perpendicular to the axis. Its length is $4a$ for a parabola and $2b^2/a$ for both an ellipse and a hyperbola.
• Tangent: A straight line that touches a conic at one point, usually without cutting it locally.
• Normal: The line perpendicular to the tangent at the point of contact.
• Asymptote: A line that a hyperbola approaches indefinitely but does not generally meet at finite distance.
• Parabola, vertex at the origin:

$$y^2=4ax$$ opens right; $$y^2=-4ax$$ opens left; $$x^2=4ay$$ opens upward; $$x^2=-4ay$$ opens downward.

• Parabola $y^2=4ax$: Focus $(a,0)$, directrix $x=-a$, axis the $x$-axis, vertex $(0,0)$, and latus rectum length $4a$.
• Parabola $x^2=4ay$: Focus $(0,a)$, directrix $y=-a$, axis the $y$-axis, vertex $(0,0)$, and latus rectum length $4a$.
• Shifted parabola:

$$(y-k{)}^2=4a(x-h)$$ has vertex $(h,k)$, focus $(h+a,k)$, and directrix $x=h-a$.

• Shifted vertical parabola:

$$(x-h{)}^2=4a(y-k)$$ has vertex $(h,k)$, focus $(h,k+a)$, and directrix $y=k-a$.

• Ellipse, major axis along the $x$-axis:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b>0.$$ Its vertices are $(\pm a,0)$, co-vertices are $(0,\pm b)$, foci are $(\pm c,0)$, where $$c^2=a^2-b^2,$$ eccentricity is $$e=\frac{c}{a},$$ and directrices are $$x=\pm \frac{a}{e}=\pm \frac{a^2}{c}.$$

• Ellipse, major axis along the $y$-axis:

$$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1,$$ with foci $(0,\pm c)$, where $$c^2=a^2-b^2.$$

• Ellipse dimensions and area: Major axis $2a$, minor axis $2b$, each latus rectum $2b^2/a$, and area

$$\pi ab.$$

• Ellipse focal property: The sum of the distances from any point on the ellipse to its two foci is

$$2a.$$

• Hyperbola, transverse axis along the $x$-axis:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$$ Its vertices are $(\pm a,0)$, foci are $(\pm c,0)$, where $$c^2=a^2+b^2,$$ eccentricity is $$e=\frac{c}{a},$$ and directrices are $$x=\pm \frac{a}{e}=\pm \frac{a^2}{c}.$$ Its asymptotes are $$y=\pm \frac{b}{a}x.$$

• Hyperbola, transverse axis along the $y$-axis:

$$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1,$$ with asymptotes $$y=\pm \frac{a}{b}x.$$

• Hyperbola dimensions: Transverse axis $2a$, conjugate axis $2b$, and each latus rectum $2b^2/a$.
• Hyperbola focal property: The absolute difference of the distances from any point on the hyperbola to its two foci is

$$2a.$$

• Parameter relations: For an ellipse,

$$a^2=b^2+c^2;$$ for a hyperbola, $$c^2=a^2+b^2.$$

• Circle: A circle is a special ellipse in which $a=b$, so its eccentricity is $e=0$.
• Focal property of a parabola: A ray parallel to the axis reflects through the focus.
• Second-degree classification: For

$$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,$$ the quantity $B^2-4AC$ is negative for an ellipse-type conic, zero for a parabola-type conic, and positive for a hyperbola-type conic, subject to non-degeneracy conditions.

Worked Methods

1. Classifying a conic using eccentricity

• Use the focus–directrix ratio

$$\frac{PF}{PM}=e.$$

• Compare $e$ with $1$:

- $e=1$: parabola. - $0<e<1$: ellipse. - $e>1$: hyperbola.

• Note that a circle is the special ellipse with $e=0$.

2. Reading the orientation and elements of a standard parabola

• Identify whether the squared term is $x^2$ or $y^2$.
• Identify the sign and compare the equation with the standard forms:

$$y^2=4ax,y^2=-4ax,x^2=4ay,x^2=-4ay.$$

• Determine the opening direction:

- $y^2=4ax$: right. - $y^2=-4ax$: left. - $x^2=4ay$: upward. - $x^2=-4ay$: downward.

• For $y^2=4ax$, record:

- Vertex $(0,0)$. - Focus $(a,0)$. - Directrix $x=-a$. - Axis: $x$-axis. - Latus rectum length $4a$.

• For $x^2=4ay$, record:

- Vertex $(0,0)$. - Focus $(0,a)$. - Directrix $y=-a$. - Axis: $y$-axis. - Latus rectum length $4a$.

3. Reading the elements of a shifted parabola

• Compare the equation with

$$(y-k{)}^2=4a(x-h)$$ or $$(x-h{)}^2=4a(y-k).$$

• Read the vertex directly as $(h,k)$.
• For

$$(y-k{)}^2=4a(x-h),$$ use focus $(h+a,k)$ and directrix $x=h-a$.

• For

$$(x-h{)}^2=4a(y-k),$$ use focus $(h,k+a)$ and directrix $y=k-a$.

• Translation changes the vertex or centre position but preserves the basic shape and properties of the conic.

4. Determining the elements of an ellipse

• Identify the larger denominator in

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b>0.$$

• If $a^2$ is under $x^2$, the major axis is along the $x$-axis:

- Vertices $(\pm a,0)$. - Co-vertices $(0,\pm b)$. - Foci $(\pm c,0)$.

• Calculate $c$ using

$$c^2=a^2-b^2.$$

• Calculate eccentricity using

$$e=\frac{c}{a}.$$

• State the directrices:

$$x=\pm \frac{a}{e}=\pm \frac{a^2}{c}.$$

• If the major axis is along the $y$-axis, use

$$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1,$$ with foci $(0,\pm c)$ and $c^2=a^2-b^2$.

• Use the remaining formulas:

- Major axis $2a$. - Minor axis $2b$. - Latus rectum $2b^2/a$. - Area $\pi ab$.

• Apply the focal property:

$$P{F}_1+P{F}_2=2a.$$

5. Determining the elements of a hyperbola

• Identify the positive squared term in the standard equation.
• For

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$$ the transverse axis is along the $x$-axis.

• Record the vertices $(\pm a,0)$.
• Calculate the focal distance using

$$c^2=a^2+b^2.$$

• Record the foci $(\pm c,0)$, eccentricity $e=c/a$, and directrices

$$x=\pm \frac{a}{e}=\pm \frac{a^2}{c}.$$

• Determine the asymptotes:

$$y=\pm \frac{b}{a}x.$$

• For

$$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1,$$ the transverse axis is along the $y$-axis and the asymptotes are $$y=\pm \frac{a}{b}x.$$

• Use the dimensions:

- Transverse axis $2a$. - Conjugate axis $2b$. - Latus rectum $2b^2/a$.

• Apply the focal property:

$$|P{F}_1-P{F}_2|=2a.$$

6. Classifying a second-degree equation

• Write the equation in the form

$$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0.$$

• Calculate

$$B^2-4AC.$$

• Subject to non-degeneracy conditions, classify it as:

- Negative: ellipse-type conic. - Zero: parabola-type conic. - Positive: hyperbola-type conic.

Where It Goes Wrong

• Confusing the focus–directrix ratio $PF/PM=e$ with an unrestricted distance statement; $M$ must be the foot of the perpendicular from $P$ to the directrix.
• Reversing the opening direction of $y^2=4ax$ and $x^2=4ay$; the squared variable determines the axis, while the sign determines the direction.
• Using $c^2=a^2+b^2$ for an ellipse instead of the required relation $c^2=a^2-b^2$; the relations are different for ellipses and hyperbolas.
• Forgetting that the directrices of the standard ellipse and hyperbola are $x=\pm a/e=\pm a^2/c$, with corresponding vertical forms when the principal axis is along the $y$-axis.
• Interchanging the hyperbola asymptotes: for $x^2/a^2-y^2/b^2=1$, they are $y=\pm (b/a)x$, whereas for $y^2/a^2-x^2/b^2=1$, they are $y=\pm (a/b)x$.
• Applying the classification test $B^2-4AC$ without observing the stated non-degeneracy conditions.

What Gets Asked

• Define a conic section, focus, directrix, eccentricity, vertex, axis, latus rectum, tangent, normal, or asymptote.
• State and use the focus–directrix definition

$$PF/PM=e$$ to classify a conic.

• Identify a parabola from its equation, state its opening direction, and determine its vertex, focus, directrix, axis, or latus rectum.
• Determine the elements of a shifted parabola from

$$(y-k{)}^2=4a(x-h)\text{or}(x-h{)}^2=4a(y-k).$$

• Find the vertices, co-vertices, foci, eccentricity, directrices, axes, latus rectum, or area of an ellipse.
• Use $a^2=b^2+c^2$ and the ellipse focal-distance sum $2a$.
• Find the vertices, foci, eccentricity, directrices, axes, latus rectum, or asymptotes of a hyperbola.
• Use $c^2=a^2+b^2$ and the hyperbola focal-distance difference $2a$.
• Explain the special case in which an ellipse becomes a circle.
• State the focal property of a parabola involving reflection of a ray parallel to its axis through its focus.
• Classify

$$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$$ using $B^2-4AC$.