Conic Sections for FSc and A Level

Slice a double cone with a flat plane and, depending on the angle of the cut, you get a circle, an ellipse, a parabola or a hyperbola. These four curves, the conic sections, describe planetary orbits, satellite dishes, bridge arches and the paths of thrown objects. In FSc they form a full chapter of analytic geometry, and A Level students meet them through coordinate geometry and parametric equations. This guide covers the standard forms, the key features of each curve, and worked examples of the questions you are most likely to face.

One Definition for All Conics

Every conic can be defined using a fixed point (the focus), a fixed line (the directrix) and a constant ratio called the eccentricity \( e \). A conic is the set of points whose distance from the focus divided by its distance from the directrix always equals \( e \).

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

Keeping this single idea in mind makes the four curves feel like members of one family.

The Circle

The circle with centre \( (h, k) \) and radius \( r \) is

$$ (x - h)^2 + (y - k)^2 = r^2 $$

The general form is \( x^2 + y^2 + 2gx + 2fy + c = 0 \), with centre \( (-g, -f) \) and radius \( \sqrt{g^2 + f^2 - c} \). Notice that a circle's equation always has equal coefficients of \( x^2 \) and \( y^2 \) and no \( xy \) term.

The tangent to \( x^2 + y^2 = r^2 \) at the point \( (x_1, y_1) \) on the circle is \( x x_1 + y y_1 = r^2 \).

Worked Example 1: Centre and Radius

Find the centre and radius of \( x^2 + y^2 - 6x + 4y - 12 = 0 \).

Step 1. Group the \( x \) and \( y \) terms and complete the square:

$$ (x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4 $$

Step 2. Simplify:

$$ (x - 3)^2 + (y + 2)^2 = 25 $$

Step 3. Read off the centre \( (3, -2) \) and radius \( r = \sqrt{25} = 5 \).

Check with the formula. Here \( 2g = -6 \), \( 2f = 4 \), \( c = -12 \), so \( g = -3 \), \( f = 2 \). The centre is \( (3, -2) \) and \( r = \sqrt{9 + 4 + 12} = 5 \). Both methods agree.

Worked Example 2: A Tangent to a Circle

Find the tangent to \( x^2 + y^2 = 25 \) at \( (3, 4) \).

Step 1. Check the point lies on the circle: \( 9 + 16 = 25 \). Yes.

Step 2. Use \( x x_1 + y y_1 = r^2 \): \( 3x + 4y = 25 \).

Check. The radius to \( (3, 4) \) has slope \( \tfrac{4}{3} \). The tangent \( 3x + 4y = 25 \) has slope \( -\tfrac{3}{4} \). Their product is \( -1 \), so they are perpendicular, as a tangent and radius must be.

The Parabola

The standard parabola opening to the right is

$$ y^2 = 4ax $$

with vertex at the origin, focus \( (a, 0) \), directrix \( x = -a \), axis along the \( x \)-axis, and latus rectum (the chord through the focus perpendicular to the axis) of length \( 4a \). The other standard forms are \( y^2 = -4ax \), \( x^2 = 4ay \) and \( x^2 = -4ay \).

Worked Example 3: Features of a Parabola

For \( y^2 = 12x \), find the focus, directrix and length of the latus rectum.

Step 1. Compare with \( y^2 = 4ax \): \( 4a = 12 \), so \( a = 3 \).

Step 2. Focus \( (3, 0) \); directrix \( x = -3 \); latus rectum length \( 4a = 12 \).

Check. The point \( (3, 6) \) is on the curve, since \( 36 = 12 \times 3 \). Its distance to the focus is 6, and its distance to the directrix is \( 3 - (-3) = 6 \). Equal distances, so \( e = 1 \), exactly as a parabola requires.

A Level students often use the parametric form \( x = at^2, \; y = 2at \), which for this parabola is \( x = 3t^2, \; y = 6t \).

The Ellipse

The standard ellipse is

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

Its major axis has length \( 2a \), minor axis \( 2b \), and foci at \( (\pm c, 0) \) where

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

The directrices are \( x = \pm \tfrac{a}{e} \). An ellipse has the beautiful property that the sum of the distances from any point on it to the two foci is always \( 2a \). The parametric form is \( x = a\cos\theta, \; y = b\sin\theta \).

Worked Example 4: Foci and Eccentricity of an Ellipse

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

Step 1. \( a^2 = 25 \), so \( a = 5 \); \( b^2 = 9 \), so \( b = 3 \).

Step 2. \( c^2 = 25 - 9 = 16 \), so \( c = 4 \). Foci: \( (\pm 4, 0) \).

Step 3. \( e = \tfrac{4}{5} = 0.8 \). This is less than 1, as it should be for an ellipse.

Step 4. Directrices: \( x = \pm \tfrac{5}{0.8} = \pm 6.25 \).

Check. The point \( (0, 3) \) lies on the ellipse. Its distance to each focus is \( \sqrt{16 + 9} = 5 \), and \( 5 + 5 = 10 = 2a \). Correct.

The Hyperbola

The standard hyperbola is

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

with vertices \( (\pm a, 0) \), foci \( (\pm c, 0) \) where \( c^2 = a^2 + b^2 \) (note the plus sign), eccentricity \( e = \tfrac{c}{a} > 1 \), and asymptotes

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

For a hyperbola, the difference of the distances from any point to the two foci is always \( 2a \).

Worked Example 5: A Hyperbola

For \( \dfrac{x^2}{16} - \dfrac{y^2}{9} = 1 \): \( a = 4 \), \( b = 3 \), \( c^2 = 16 + 9 = 25 \), so \( c = 5 \). The foci are \( (\pm 5, 0) \), the eccentricity is \( \tfrac{5}{4} = 1.25 \), and the asymptotes are \( y = \pm \tfrac{3}{4}x \).

Recognising a Conic From Its Equation

For an equation \( Ax^2 + Cy^2 + Dx + Ey + F = 0 \) with no \( xy \) term:

• \( A = C \): circle
• \( A \) or \( C \) is zero (but not both): parabola
• \( A \) and \( C \) have the same sign but differ: ellipse
• \( A \) and \( C \) have opposite signs: hyperbola

Then complete the square to put the equation into standard form.

Did You Know?

• Apollonius of Perga, working around 200 BC, wrote eight books on conics and gave us the names ellipse, parabola and hyperbola.
• In 1609 Johannes Kepler showed that planets move in ellipses with the Sun at one focus, turning ancient geometry into modern astronomy.
• A satellite dish is a parabola rotated about its axis: incoming parallel signals reflect off the dish and meet at the focus, where the receiver sits.

Key Takeaways

• Eccentricity classifies conics: \( e = 0 \) circle, \( e < 1 \) ellipse, \( e = 1 \) parabola, \( e > 1 \) hyperbola.
• Completing the square is the main skill for finding centres, vertices and radii.
• For an ellipse \( c^2 = a^2 - b^2 \); for a hyperbola \( c^2 = a^2 + b^2 \). Do not mix them up.
• Check answers using the defining property: equal distances for a parabola, constant sum for an ellipse, constant difference for a hyperbola.
• Parametric forms are especially useful at A Level for tangents and normals.

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