Conic Sections

The One Thing

Conic sections are curves formed by intersecting a plane with a double cone. Their geometric definitions translate into standard equations, which determine properties such as position, orientation, vertices, foci, directrices, axes, eccentricity, latus rectum, tangents, normals, and asymptotes.

Definitions and Results

• Conic section: A curve obtained by intersecting a plane with a double cone.

• Circle: The set of all points in a plane at a fixed distance from a fixed point called the centre. A circle with centre $(h,k)$ and radius $r$ has equation

$$(x-h{)}^2+(y-k{)}^2=r^2.$$ The circle is a special ellipse whose two foci coincide at the centre and whose eccentricity is $0$.

• Parabola: The set of points equidistant from a fixed point called the focus and a fixed line called the directrix. Its eccentricity is $1$.

• Ellipse: The set of points for which the sum of the distances from two fixed points, called foci, is constant. Its eccentricity is less than $1$.

• Hyperbola: The set of points for which the absolute difference of the distances from two fixed points, called foci, is constant. Its eccentricity is greater than $1$.

• Focus: A fixed point used to define a parabola, ellipse, or hyperbola.

• Directrix: A fixed line used with a focus to define a parabola or to describe the eccentricity of a conic.

• Eccentricity: The ratio

$$e=\frac{\text{distance from a point to the focus}}{\text{perpendicular distance from the point to the directrix}}.$$ It is $1$ for a parabola, less than $1$ for an ellipse, and greater than $1$ for a hyperbola.

• Vertex: A point where the conic meets its axis or where the curve changes direction.

• Axis: A line of symmetry or a principal line associated with a conic.

• Latus rectum: A chord of a conic passing through its focus and perpendicular to its axis.

• Chord of contact: The line joining the points of contact of tangents drawn from an external point to a conic.

• Tangent: A line that touches a conic at exactly one point, subject to the usual non-degenerate conditions.

• Normal: A line perpendicular to the tangent at the point of contact.

• General equation of a circle:

$$x^2+y^2+2gx+2fy+c=0.$$ Its centre is $(-g,-f)$, and its radius is $$\sqrt{g^2+f^2-c}.$$

• Circle with a given diameter: If the endpoints of the diameter are $(x_1,y_1)$ and $(x_2,y_2)$, its equation is

$$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0.$$

• Standard parabola:

$$y^2=4ax.$$ The vertex is $(0,0)$, the focus is $(a,0)$, the directrix is $x=-a$, the axis is the $x$-axis, and the length of the latus rectum is $4a$.

• Standard vertical parabola:

$$x^2=4ay.$$ The focus is $(0,a)$, the directrix is $y=-a$, the axis is the $y$-axis, and the length of the latus rectum is $4a$.

• Reflected parabolas: The equation $y^2=-4ax$ represents a parabola opening left, while $x^2=-4ay$ represents a parabola opening downward.

• Standard ellipse:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b.$$ The centre is $(0,0)$, the vertices are $(\pm a,0)$, the foci are $(\pm c,0)$, and $$c^2=a^2-b^2.$$ The major axis has length $2a$, the minor axis has length $2b$, the eccentricity is $$e=\frac{c}{a},$$ and the latus rectum has length $$\frac{2b^2}{a}.$$

• Vertical ellipse:

$$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1,a>b.$$ The major axis is vertical and the foci are $(0,\pm c)$, where $$c^2=a^2-b^2.$$

• Standard hyperbola:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$$ The centre is $(0,0)$, the vertices are $(\pm a,0)$, the foci are $(\pm c,0)$, and $$c^2=a^2+b^2.$$ The transverse axis has length $2a$, the conjugate axis has length $2b$, the eccentricity is $$e=\frac{c}{a},$$ and the latus rectum has length $$\frac{2b^2}{a}.$$ Its asymptotes are $$y=\pm \frac{b}{a}x.$$

• Vertical hyperbola:

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

• Translated parabola:

$$(y-k{)}^2=4a(x-h)$$ or $$(x-h{)}^2=4a(y-k).$$ In either form, the vertex is $(h,k)$.

• Translated ellipse:

$$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1.$$

• Translated hyperbola:

$$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$$ or $$\frac{(y-k{)}^2}{a^2}-\frac{(x-h{)}^2}{b^2}=1.$$

• Focus–directrix relation: For a conic defined by a focus and directrix,

$$\frac{\text{distance from focus}}{\text{distance from directrix}}=e.$$

• Classification by discriminant: For

$$A{x}^2+2Hxy+B{y}^2+2Gx+2Fy+C=0,$$ the quantity $H^2-AB$ helps classify the conic, subject to non-degeneracy conditions: $$H^2-AB<0$$ generally indicates an ellipse or circle, $$H^2-AB=0$$ indicates a parabola, and $$H^2-AB>0$$ indicates a hyperbola.

• Tangent to a parabola: The tangent to

$$y^2=4ax$$ at $(x_1,y_1)$ is $$y{y}_1=2a(x+x_1).$$

• Tangent to an ellipse: The tangent to

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ at $(x_1,y_1)$ is $$\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=1.$$

• Tangent to a hyperbola: The tangent to

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ at $(x_1,y_1)$ is $$\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1.$$

• Line equations: The slope form is

$$y=mx+c,$$ and the point-slope form is $$y-y_1=m(x-x_1).$$ These forms are frequently used to find tangents and normals.

Worked Methods

Identifying a conic from its geometric definition

• Use the defining distance relationship:

- one fixed distance from a centre: circle; - equal distances from a focus and directrix: parabola; - constant sum of distances from two foci: ellipse; - constant absolute difference of distances from two foci: hyperbola.

• Use the eccentricity when a focus–directrix description is given:

e=1 \text{ for a parabola},\qquad e<1 \text{ for an ellipse},\qquad e>1 \text{ for a hyperbola}. 3. Recognise that coordinate geometry converts the relevant distance condition into an equation. ### Extracting properties from a standard parabola For \[ y^2=4ax:

• Identify the vertex as $(0,0)$.
• Identify the axis as the $x$-axis.
• Read the focus as $(a,0)$.
• Read the directrix as $x=-a$.
• State that the parabola opens to the right when $a>0$.
• Use the latus rectum length $4a$.

For $$x^2=4ay:$$

• Identify the focus as $(0,a)$.
• Identify the directrix as $y=-a$.
• Identify the axis as the $y$-axis.
• Use the latus rectum length $4a$.

For $$y^2=-4ax,$$ the parabola opens left. For $$x^2=-4ay,$$ it opens downward.

For a translated parabola:

• Compare the equation with

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

• Read the vertex as $(h,k)$.
• Determine the orientation from the sign of $a$ and the variable appearing linearly.

Extracting properties from a standard ellipse

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

• Identify the centre as $(0,0)$.
• Since the larger denominator is under $x^2$, identify the major axis as horizontal.
• Identify the vertices as $(\pm a,0)$.
• Calculate the focal distance using

$$c=\sqrt{a^2-b^2}.$$

• Identify the foci as $(\pm c,0)$.
• Calculate the major axis length $2a$, minor axis length $2b$, eccentricity $e=c/a$, and latus rectum length $2b^2/a$.

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

• Identify the major axis as vertical.
• Calculate

$$c=\sqrt{a^2-b^2}.$$

• Identify the foci as $(0,\pm c)$.

For a translated ellipse:

• Compare the equation with

$$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1.$$

• Identify the centre as $(h,k)$.
• Determine the orientation by locating the larger denominator.

Extracting properties from a standard hyperbola

For $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1:$$

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

$$c=\sqrt{a^2+b^2}.$$

• Identify the foci as $(\pm c,0)$.
• State that the transverse axis has length $2a$ and the conjugate axis has length $2b$.
• Calculate the eccentricity $e=c/a$ and latus rectum length $2b^2/a$.
• Find the asymptotes from

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

For $$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1:$$

• Identify the transverse direction as vertical.

2.