Conic Sections: Parabola, Ellipse and Hyperbola

Conic sections are the curves you get when a plane slices through a double cone: circles, ellipses, parabolas and hyperbolas. They appear in FSc Part 2, A Level Further Maths and first-year university courses, and they describe real things such as satellite dishes, planetary orbits and the paths of projectiles. This guide gives you the standard equations, the key features of each curve and worked examples that show exactly how to find them.

How the Curves Arise

Picture two cones joined tip to tip. Cutting them with a plane gives:

• Circle: the plane is perpendicular to the axis.
• Ellipse: the plane is tilted but cuts only one cone in a closed curve.
• Parabola: the plane is parallel to the side of the cone.
• Hyperbola: the plane cuts both cones, giving two separate branches.

The Unifying Idea: Focus, Directrix and Eccentricity

Every conic (except the circle) can be defined as the set of points \( P \) for which

$$ \frac{\text{distance from } P \text{ to the focus}}{\text{distance from } P \text{ to the directrix}} = e $$

where \( e \) is the eccentricity, a constant:

• \( e = 0 \): circle
• \( 0 < e < 1 \): ellipse
• \( e = 1 \): parabola
• \( e > 1 \): hyperbola

The Parabola

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix), so \( e = 1 \).

Standard form (opening to the right):

$$ y^2 = 4ax $$

The other orientations are \( y^2 = -4ax \) (opens left), \( x^2 = 4ay \) (opens up) and \( x^2 = -4ay \) (opens down).

Worked Example 1: Parabola Features

Question: Find the focus, directrix and length of the latus rectum of \( y^2 = 12x \).

Step 1: Compare with \( y^2 = 4ax \).

$$ 4a = 12 \Rightarrow a = 3 $$

Step 2: Read off the features.

• Focus: \( (3, 0) \)
• Directrix: \( x = -3 \)
• Latus rectum: \( 4a = 12 \)

Check: The point \( (3, 6) \) lies on the curve since \( 6^2 = 36 = 12 \times 3 \). Its distance to the focus is \( 6 \), and its distance to the directrix is \( 3 - (-3) = 6 \). Equal, as the definition requires.

The Ellipse

An ellipse is the set of points where the sum of the distances to two foci is constant, equal to \( 2a \).

Standard form (major axis along the \( x \)-axis, \( a > b \)):

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

The foci are at \( (\pm c, 0) \), where

$$ c^2 = a^2 - b^2, \qquad e = \frac{c}{a} $$

The vertices are \( (\pm a, 0) \), the length of the major axis is \( 2a \) and the minor axis is \( 2b \). The directrices are \( x = \pm \dfrac{a}{e} \). A circle is the special case \( a = b \), giving \( c = 0 \) and \( e = 0 \).

Worked Example 2: Ellipse Features

Question: For \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 \), find the vertices, foci and eccentricity.

Step 1: Identify \( a \) and \( b \).

$$ a^2 = 25 \Rightarrow a = 5, \qquad b^2 = 9 \Rightarrow b = 3 $$

Step 2: Find \( c \).

$$ c^2 = 25 - 9 = 16 \Rightarrow c = 4 $$

Step 3: State the features.

• Vertices: \( (\pm 5, 0) \); co-vertices: \( (0, \pm 3) \)
• Foci: \( (\pm 4, 0) \)
• Eccentricity: \( e = \dfrac{4}{5} = 0.8 \)

Check: For the co-vertex \( (0, 3) \), each focus is \( \sqrt{4^2 + 3^2} = 5 \) away, so the sum is \( 10 = 2a \). Correct.

The Hyperbola

A hyperbola is the set of points where the difference of the distances to two foci is constant, equal to \( 2a \).

Standard form (opening left and right):

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

Now the relationship changes sign:

$$ c^2 = a^2 + b^2, \qquad e = \frac{c}{a} > 1 $$

The vertices are \( (\pm a, 0) \), the foci are \( (\pm c, 0) \), and the curve approaches two asymptotes:

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

Worked Example 3: Hyperbola Features

Question: For \( \dfrac{x^2}{16} - \dfrac{y^2}{9} = 1 \), find the vertices, foci, eccentricity and asymptotes.

Step 1: Identify \( a \) and \( b \). \( a = 4 \), \( b = 3 \).

Step 2: Find \( c \).

$$ c^2 = 16 + 9 = 25 \Rightarrow c = 5 $$

Step 3: State the features.

• Vertices: \( (\pm 4, 0) \)
• Foci: \( (\pm 5, 0) \)
• Eccentricity: \( e = \dfrac54 = 1.25 \)
• Asymptotes: \( y = \pm \dfrac{3}{4}x \)

Worked Example 4: Completing the Square

Exam equations are often not in standard form.

Question: Identify the conic \( 4x^2 + 9y^2 - 16x + 18y - 11 = 0 \) and find its centre and foci.

Step 1: Group and factor.

$$ 4(x^2 - 4x) + 9(y^2 + 2y) = 11 $$

Step 2: Complete each square.

$$ 4\left[(x - 2)^2 - 4\right] + 9\left[(y + 1)^2 - 1\right] = 11 $$

$$ 4(x - 2)^2 + 9(y + 1)^2 = 11 + 16 + 9 = 36 $$

Step 3: Divide by 36.

$$ \frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1 $$

Step 4: Interpret. Both squared terms are positive with different denominators, so this is an ellipse with centre \( (2, -1) \), \( a = 3 \) and \( b = 2 \).

$$ c = \sqrt{9 - 4} = \sqrt{5} $$

The foci are \( \left(2 \pm \sqrt{5},\ -1\right) \).

Quick Identification from \( Ax^2 + Cy^2 + \dots = 0 \)

• \( A = C \): circle
• \( A \) and \( C \) same sign, different values: ellipse
• \( A \) and \( C \) opposite signs: hyperbola
• Only one of \( A \) or \( C \) is non-zero: parabola

Common Mistakes

• Using \( c^2 = a^2 - b^2 \) for a hyperbola. For a hyperbola, \( c^2 = a^2 + b^2 \).
• Assuming \( a \) is always under \( x^2 \). In an ellipse, \( a \) is the larger semi-axis; in a hyperbola, \( a \) is under the positive term.
• Forgetting to shift foci and vertices by the centre after completing the square.
• Mixing up \( 4a \) and \( a \) in parabola questions.

Key Takeaways

• Eccentricity classifies conics: \( e = 0 \) circle, \( e < 1 \) ellipse, \( e = 1 \) parabola, \( e > 1 \) hyperbola.
• Parabola \( y^2 = 4ax \): focus \( (a, 0) \), directrix \( x = -a \).
• Ellipse: \( c^2 = a^2 - b^2 \); the sum of focal distances is \( 2a \).
• Hyperbola: \( c^2 = a^2 + b^2 \); asymptotes \( y = \pm \tfrac{b}{a}x \).
• Complete the square to convert a general equation to standard form.

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