Chapter 06 — Coordinate Geometry: Part 8 — Conic Sections: General Review and Mixed

MATHEMATICS HIPHOP ACADEMY

CHAPTER 06: COORDINATE GEOMETRY

PART 8 — CONIC SECTIONS: GENERAL REVIEW AND MIXED EXAMINATION PRACTICE

A-Level Pure Mathematics Paper 1

Learning objectives

By the end of this lesson, you should be able to:

• Recognise the four main conic sections.
• Distinguish between their standard equations and geometrical properties.
• Identify centres, vertices, foci, directrices and asymptotes where applicable.
• Use eccentricity to distinguish between conic sections.
• Convert equations into standard form by completing the square.
• Apply tangent and normal methods to conic sections.
• Solve mixed examination questions involving conics.
• Check solutions and interpret their geometrical meaning.

1. What Are Conic Sections?

Conic sections are curves obtained by intersecting a plane with a double cone. The four main types studied in coordinate geometry are:

1. Parabola
2. Ellipse
3. Hyperbola
4. Rectangular hyperbola

Each curve has its own defining properties, standard equations and geometrical features.

1.1 The four conic sections

The parabola

A parabola is the locus of a point whose distance from a fixed point called the focus is equal to its perpendicular distance from a fixed line called the directrix.

Standard example: y² = 4ax.

The ellipse

An ellipse is the locus of a point for which the sum of its distances from two fixed points called the foci is constant.

Standard example: x²/a² + y²/b² = 1, where a is greater than b for a horizontal major axis.

The hyperbola

A hyperbola is the locus of a point for which the absolute difference of its distances from two fixed points called the foci is constant.

Standard example: x²/a² − y²/b² = 1.

The rectangular hyperbola

A rectangular hyperbola is a special hyperbola whose asymptotes are perpendicular to one another.

A common form is xy = k, where k is a non-zero constant.

2. Comparing the Standard Equations

The first step in solving a conic-section problem is recognising the equation.

Conic	Standard equation	Main feature
Parabola	y² = 4ax	One focus and one directrix
Ellipse	x²/a² + y²/b² = 1	Closed curve with two foci
Hyperbola	x²/a² − y²/b² = 1	Two branches and two asymptotes
Rectangular hyperbola	xy = k	Perpendicular asymptotes

Important observation: The signs and arrangement of the squared terms help identify the curve.

• One squared variable and one linear variable commonly indicate a parabola.
• Two positive squared terms with equal signs commonly indicate an ellipse.
• Two squared terms with opposite signs commonly indicate a hyperbola.
• A product of the variables, such as xy = k, can represent a rectangular hyperbola.

3. Eccentricity: Comparing the Curves

Eccentricity is a numerical measure used to describe the shape of a conic section.

Conic	Eccentricity
Circle	e = 0
Ellipse	0 < e < 1
Parabola	e = 1
Hyperbola	e > 1

The circle is a special case of an ellipse. Its two foci coincide at the centre.

3.1 Eccentricity formulae

Ellipse:

c² = a² − b², and e = c/a.

Hyperbola:

c² = a² + b², and e = c/a.

Parabola:

The focus-directrix definition gives eccentricity e = 1.

Remember that a, b and c have different meanings depending on the conic. Always identify the curve before selecting a formula.

4. Essential Formula Review

4.1 Parabola

For y² = 4ax:

• Vertex: (0, 0)
• Focus: (a, 0)
• Directrix: x = −a
• Axis: y = 0
• Latus rectum length: 4a
• Parametric point: (at², 2at)
• Tangent at parameter t: ty = x + at²

For x² = 4ay, the parabola opens upwards when a is positive.

• Focus: (0, a)
• Directrix: y = −a
• Axis: x = 0

4.2 Ellipse

For a horizontal ellipse:

x²/a² + y²/b² = 1, where a > b.

• Centre: (0, 0)
• Vertices: (±a, 0)
• Co-vertices: (0, ±b)
• Foci: (±c, 0)
• c² = a² − b²
• e = c/a
• Parametric point: (a cos θ, b sin θ)
• Tangent: xx₁/a² + yy₁/b² = 1

4.3 Hyperbola

For a horizontal hyperbola:

x²/a² − y²/b² = 1.

• Centre: (0, 0)
• Vertices: (±a, 0)
• Foci: (±c, 0)
• c² = a² + b²
• e = c/a
• Asymptotes: y = ±(b/a)x
• Latus rectum length: 2b²/a
• Tangent: xx₁/a² − yy₁/b² = 1

4.4 Rectangular hyperbola

For xy = k:

• The coordinate axes are asymptotes.
• The curve does not cross either coordinate axis when k is non-zero.
• The branches lie in the first and third quadrants if k is positive.
• The branches lie in the second and fourth quadrants if k is negative.

5. Completing the Square: A General Skill

Many examination questions give a conic in expanded form. Completing the square allows us to identify its centre and standard equation.

Consider:

x²/4 + y²/9 − x + 2y − 2 = 0

Step 1: Multiply through by 36.

9x² + 4y² − 36x + 72y − 72 = 0.

Step 2: Group the variables.

9(x² − 4x) + 4(y² + 18y) = 72.

Step 3: Complete the square.

x² − 4x = (x − 2)² − 4.

y² + 18y = (y + 9)² − 81.

Step 4: Substitute.

9[(x − 2)² − 4] + 4[(y + 9)² − 81] = 72.

Step 5: Simplify.

9(x − 2)² − 36 + 4(y + 9)² − 324 = 72.

9(x − 2)² + 4(y + 9)² = 432.

Step 6: Divide through by 432.

(x − 2)²/48 + (y + 9)²/108 = 1

This is an ellipse with centre (2, −9). Its major axis is vertical because the larger denominator is under the y-term.

6. Tangents and Normals: Comparing Methods

A tangent touches a curve at a point. A normal passes through the same point at right angles to the tangent.

6.1 Tangent formulae

Parabola y² = 4ax:

At (at², 2at), the tangent is:

ty = x + at².

Ellipse x²/a² + y²/b² = 1:

At (x₁, y₁), the tangent is:

xx₁/a² + yy₁/b² = 1.

Horizontal hyperbola x²/a² − y²/b² = 1:

At (x₁, y₁), the tangent is:

xx₁/a² − yy₁/b² = 1.

6.2 Using differentiation

For a curve expressed implicitly as F(x, y) = 0, differentiate with respect to x, treating y as a function of x.

For example, for the ellipse:

x²/a² + y²/b² = 1

Differentiating:

2x/a² + (2y/b²)(dy/dx) = 0.

Therefore:

dy/dx = −b²x/(a²y)

At a point where the tangent gradient is m, the normal gradient is −1/m, provided m is non-zero. If the tangent is horizontal or vertical, determine the perpendicular line directly.

Examination reminder: Always verify that a stated point lies on the curve before applying a tangent formula.

7. Worked Mixed Examples

Example 1: Identifying a conic from its equation

Identify the conic represented by:

4x² + 9y² = 36.

Solution

Divide through by 36:

x²/9 + y²/4 = 1.

Both squared terms are positive and have different denominators. Therefore, the curve is an ellipse.

a² = 9 and b² = 4.

Centre: (0, 0).

Since the larger denominator is under x², the major axis is horizontal.

Example 2: Finding the focus and directrix of a parabola

Find the focus and directrix of:

y² = 20x.

Solution

Compare with y² = 4ax.

4a = 20, so a = 5.

Focus: (5, 0).

Directrix: x = −5.

Vertex: (0, 0).

Example 3: Finding the foci of an ellipse

Find the foci and eccentricity of:

x²/100 + y²/64 = 1.

Solution

a² = 100 and b² = 64.

c² = a² − b² = 100 − 64 = 36.

Therefore c = 6.

Foci: (−6, 0) and (6, 0).

e = c/a = 6/10 = 3/5.

Example 4: Finding the asymptotes of a hyperbola

Find the asymptotes of:

x²/25 − y²/16 = 1.

Solution

a = 5 and b = 4.

The asymptotes are:

y = ±(b/a)x.

Therefore:

y = 4x/5 and y = −4x/5.

Example 5: A rectangular hyperbola

Given xy = 16, find y when x = 4 and describe the asymptotes.

Solution

xy = 16.

When x = 4:

4y = 16.

y = 4.

The point is (4, 4).

The asymptotes are x = 0 and y = 0, which are the coordinate axes.

8. Mixed Examination Practice

Attempt every question before opening the answers. Show your working and give exact values where possible.

Exercise A: Recognition and properties

Question 1. Identify each of the following curves:

1. y² = 28x
2. x²/36 + y²/16 = 1
3. x²/9 − y²/25 = 1
4. xy = −10

Question 2. State the possible range of eccentricity for:

1. An ellipse
2. A parabola
3. A hyperbola

Question 3. Explain two differences between an ellipse and a hyperbola.

Exercise B: Parabolas

Question 1. Find the focus, directrix and latus rectum length of y² = 32x.

Question 2. Find the equation of the tangent to y² = 12x at the point (3, 6).

Question 3. A parabola has vertex at the origin and focus (0, −4). Find its equation and directrix.

Exercise C: Ellipses

Question 1. For x²/169 + y²/25 = 1, find:

1. The centre
2. The vertices and co-vertices
3. The foci
4. The eccentricity

Question 2. An ellipse has centre (0, 0), major axis length 20 and minor axis length 12. Find its equation if the major axis is horizontal.

Question 3. Find the tangent to x²/25 + y²/9 = 1 at the point (4, 9/5).

Exercise D: Hyperbolas

Question 1. For x²/64 − y²/36 = 1, find:

1. The vertices
2. The foci
3. The eccentricity
4. The asymptotes

Question 2. A hyperbola has vertices (±4, 0) and eccentricity 5/4. Find its equation.

Question 3. Find the tangent to x²/16 − y²/9 = 1 at (5, 9/4).

Exercise E: Completing the square

Question 1. Express the equation

x² + 4y² − 6x + 8y − 3 = 0

in standard form. Identify the conic and its centre.

Question 2. Express

9x² − 16y² − 54x − 64y − 127 = 0

in standard form. Find its centre and orientation.

Exercise F: Rectangular hyperbolas

Question 1. Given xy = 24, find:

1. y when x = 6
2. x when y = 3
3. The asymptotes

Question 2. Explain why the asymptotes of xy = −15 are perpendicular.

Exercise G: Longer examination questions

Question 1. An ellipse has equation:

x²/100 + y²/64 = 1.

A point P on the ellipse has coordinates (6, 32/5).

1. Verify that P lies on the ellipse.
2. Find the equation of the tangent at P.
3. Find the gradient of the tangent at P.

Question 2. A hyperbola has equation:

(x − 2)²/25 − (y + 1)²/144 = 1.

1. Find its centre.
2. Find its vertices.
3. Find its foci.
4. Find its asymptotes.

Question 3. A parabola has equation y² = 16x. A point on the parabola is given by the parameter t.

1. Write down the parametric coordinates of the point.
2. Find the tangent at the point corresponding to t = 2.
3. Find the gradient of the tangent using differentiation.

9. Answers and Worked Solutions

Open each section after attempting the questions.

Exercise A — Solutions

Question 1.

1. Parabola.
2. Ellipse.
3. Hyperbola.
4. Rectangular hyperbola.

Question 2.

1. Ellipse: 0 < e < 1.
2. Parabola: e = 1.
3. Hyperbola: e > 1.

Question 3.

• An ellipse is a closed curve, while a hyperbola has two separate branches.
• An ellipse has eccentricity less than 1, while a hyperbola has eccentricity greater than 1.
• The ellipse has a sum-of-distances definition, while the hyperbola has an absolute difference-of-distances definition.

Exercise B — Solutions

Question 1.

y² = 32x = 4(8)x, so a = 8.

• Focus: (8, 0)
• Directrix: x = −8
• Latus rectum length: 4a = 32

Question 2.

Compare y² = 12x with y² = 4ax.

4a = 12, so a = 3.

The point (3, 6) corresponds to t = 1 because (at², 2at) = (3, 6).

Tangent: ty = x + at².

Therefore: y = x + 3.

Question 3.

The focus (0, −4) means a = −4 in x² = 4ay.

Equation: x² = −16y.

Directrix: y = 4.

Exercise C — Solutions

Question 1.

a = 13 and b = 5.

c² = 169 − 25 = 144, so c = 12.

• Centre: (0, 0)
• Vertices: (±13, 0)
• Co-vertices: (0, ±5)
• Foci: (±12, 0)
• e = 12/13

Question 2.

a = 10 and b = 6.

Equation: x²/100 + y²/36 = 1.

Question 3.

First verify the point:

16/25 + (81/25)/9 = 16/25 + 9/25 = 1.

The point lies on the ellipse.

Tangent:

4x/25 + (9/5)y/9 = 1.

4x/25 + y/5 = 1.

Multiply through by 25:

4x + 5y = 25.

Exercise D — Solutions

Question 1.

a = 8, b = 6.

c² = 64 + 36 = 100, so c = 10.

• Vertices: (±8, 0)
• Foci: (±10, 0)
• e = 10/8 = 5/4
• Asymptotes: y = ±(3/4)x

Question 2.

a = 4 and e = 5/4.

Since e = c/a, c = 5.

b² = c² − a² = 25 − 16 = 9.

Equation: x²/16 − y²/9 = 1.

Question 3.

First verify:

25/16 − (81/16)/9 = 25/16 − 9/16 = 1.

The point is on the hyperbola.

Tangent:

5x/16 − (9/4)y/9 = 1.

5x/16 − y/4 = 1.

Multiply through by 16:

5x − 4y = 16.

Exercise E — Solutions

Question 1.

Group the terms:

(x² − 6x) + 4(y² + 2y) = 3.

Complete the square:

(x − 3)² − 9 + 4[(y + 1)² − 1] = 3.

(x − 3)² + 4(y + 1)² = 16.

Divide by 16:

(x − 3)²/16 + (y + 1)²/4 = 1.

It is an ellipse with centre (3, −1).

Question 2.

Group and complete the square:

9(x² − 6x) − 16(y² + 4y) = 127.

9[(x − 3)² − 9] − 16[(y + 2)² − 4] = 127.

9(x − 3)² − 16(y + 2)² = 144.

Divide by 144:

(x − 3)²/16 − (y + 2)²/9 = 1.

It is a horizontal hyperbola with centre (3, −2).

Exercise F — Solutions

Question 1.

xy = 24.

1. When x = 6, y = 24/6 = 4.
2. When y = 3, x = 24/3 = 8.
3. Asymptotes: x = 0 and y = 0.

Question 2.

The coordinate axes are the asymptotes. The x-axis and y-axis meet at a right angle, so the asymptotes are perpendicular. Therefore, xy = −15 is a rectangular hyperbola.

Exercise G — Solutions

Question 1.

Ellipse: x²/100 + y²/64 = 1.

(a) Verification

At P = (6, 32/5):

36/100 + (1024/25)/64 = 9/25 + 16/25 = 1.

Therefore, P lies on the ellipse.

(b) Tangent

xx₁/100 + yy₁/64 = 1.

6x/100 + (32/5)y/64 = 1.

3x/50 + y/10 = 1.

Multiply through by 50:

3x + 5y = 50.

(c) Gradient

Rearrange the tangent:

5y = −3x + 50.

y = −3x/5 + 10.

Gradient = −3/5.

Question 2.

Compare with:

(x − h)²/a² − (y − k)²/b² = 1.

a = 5, b = 12, h = 2 and k = −1.

• Centre: (2, −1)
• Vertices: (−3, −1) and (7, −1)

c² = 25 + 144 = 169, so c = 13.

Foci: (−11, −1) and (15, −1).

Asymptotes:

y + 1 = ±(12/5)(x − 2).

Question 3.

For y² = 16x, compare with y² = 4ax.

4a = 16, so a = 4.

(a) Parametric coordinates

(x, y) = (4t², 8t).

(b) Tangent at t = 2

Point: (4(2²), 8(2)) = (16, 16).

Tangent formula: ty = x + at².

2y = x + 16.

x − 2y + 16 = 0.

(c) Gradient

Differentiate y² = 16x:

2y(dy/dx) = 16.

dy/dx = 8/y.

At y = 16:

Gradient = 1/2.

10. Common Examination Mistakes

1. Using the wrong focal relationship. For an ellipse, c² = a² − b². For a hyperbola, c² = a² + b².
2. Confusing a and b. In an ellipse, a is associated with the larger denominator. In a hyperbola, a belongs to the positive squared term.
3. Giving the wrong orientation. The positive squared term determines the transverse axis of a hyperbola.
4. Forgetting to translate the centre. In translated equations, use (x − h) and (y − k) carefully.
5. Incorrect completing of the square. Remember to account for the coefficient outside each bracket.
6. Using a tangent formula without checking the point. Substitute the coordinates into the original curve first.
7. Confusing a tangent with a normal. Their gradients are negative reciprocals when both gradients are defined and non-zero.
8. Rounding too early. Keep exact fractions and surds until the final answer.

11. Final Conic Sections Formula Sheet

Property	Parabola	Ellipse	Hyperbola
Standard form	y² = 4ax	x²/a² + y²/b² = 1	x²/a² − y²/b² = 1
Eccentricity	1	c/a < 1	c/a > 1
Focal relation	Focus at (a, 0)	c² = a² − b²	c² = a² + b²
Asymptotes	None	None	y = ±(b/a)x

Your final revision checklist

Before moving on from conic sections, make sure you can:

• Recognise each conic from its equation.
• Sketch its general shape and identify its orientation.
• Find its centre, vertices and foci where applicable.
• Calculate eccentricity and directrices.
• Find asymptotes of hyperbolas.
• Complete the square correctly.
• Find tangents and normals.
• Use parametric equations where required.
• Explain the geometrical meaning of your answers.

MATHEMATICS HIPHOP

Making Mathematics Easier, One Concept at a Time.

End of Chapter 06 — Part 8