What this chapter is about
Conic sections are curves obtained when a plane cuts through a double-napped right circular cone at different angles. Depending on the angle and position of the cutting plane, you get a circle, an ellipse, a parabola or a hyperbola. These curves appear everywhere in nature and engineering: the path of a thrown ball is a parabola, planets move in elliptical orbits, and satellite dish reflectors use parabolic shapes to focus signals.
In this chapter, you learn the standard equations of each conic when the centre (or vertex) is at the origin and the axes lie along the coordinate axes. You study how quantities like the focus, directrix, eccentricity, latus rectum and the lengths of major and minor axes define each curve. The chapter builds directly on your knowledge of coordinate geometry from earlier classes and prepares you for calculus-based analysis of curves in later studies.
After working through this material, you should be able to identify a conic from its equation, write the equation of a conic given geometric data, and sketch each curve showing its key features.
Key ideas
- A conic section is the locus of a point whose distance from a fixed point (focus) bears a constant ratio (eccentricity, e) to its distance from a fixed line (directrix).
- For a circle, e = 0; for an ellipse, 0 < e < 1; for a parabola, e = 1; for a hyperbola, e > 1.
- A circle is the set of all points equidistant from a fixed centre; its equation with centre (h, k) and radius r is (x − h)² + (y − k)² = r².
- A parabola has one focus and one directrix; its simplest forms are y² = 4ax (opening rightward) and x² = 4ay (opening upward).
- An ellipse has two foci; its standard form with centre at origin is x²/a² + y²/b² = 1, where a > b > 0 and c² = a² − b² gives the distance from centre to each focus.
- A hyperbola has two foci and two branches; its standard form is x²/a² − y²/b² = 1, where c² = a² + b² and the asymptotes are y = ±(b/a)x.
- The latus rectum is the chord through a focus perpendicular to the principal axis; its length helps in sketching and in problems involving focal chords.
- Every conic can also be written in general second-degree form Ax² + Bxy + Cy² + Dx + Ey + F = 0; the value of B² − 4AC decides the type (circle, ellipse, parabola or hyperbola) when B = 0 reduces the analysis.
Formulas and facts to remember
- Conic: Circle · Standard equation: x² + y² = r² · Eccentricity: 0 · Foci: centre itself · Directrix: none · Latus rectum length: 2r (diameter)
- Conic: Parabola · Standard equation: y² = 4ax · Eccentricity: 1 · Foci: (a, 0) · Directrix: x = −a · Latus rectum length: 4a
- Conic: Ellipse (a > b) · Standard equation: x²/a² + y²/b² = 1 · Eccentricity: c/a, c² = a² − b² · Foci: (±c, 0) · Directrix: x = ±a/e · Latus rectum length: 2b²/a
- Conic: Hyperbola · Standard equation: x²/a² − y²/b² = 1 · Eccentricity: c/a, c² = a² + b² · Foci: (±c, 0) · Directrix: x = ±a/e · Latus rectum length: 2b²/a
- For an ellipse, the sum of distances from any point on the curve to the two foci equals 2a.
- For a hyperbola, the absolute difference of distances from any point on the curve to the two foci equals 2a.
- Asymptotes of the hyperbola x²/a² − y²/b² = 1 are y = (b/a)x and y = −(b/a)x.
Worked examples
Example 1: Finding the equation of a parabola
A parabola has its vertex at the origin and focus at (3, 0). Find its equation and the length of its latus rectum.
Solution
Since the focus lies on the positive x-axis, the parabola opens rightward. The standard form for such a parabola is y² = 4ax, where the focus is at (a, 0).
Here a = 3.
Equation: y² = 4 × 3 × x, so y² = 12x.
Latus rectum length = 4a = 4 × 3 = 12 units.
Example 2: Equation of an ellipse from given data
An ellipse has its centre at the origin, major axis along the x-axis, semi-major axis length 5 and eccentricity 3/5. Find its equation and the coordinates of the foci.
Solution
Given: a = 5, e = 3/5.
We know e = c/a, so c = ae = 5 × (3/5) = 3.
Using c² = a² − b²: 9 = 25 − b² b² = 16, hence b = 4.
Standard form: x²/25 + y²/16 = 1.
Foci are at (±c, 0) = (±3, 0).
Example 3: Identifying a hyperbola and its asymptotes
Identify the conic 9x² − 16y² = 144 and find its asymptotes.
Solution
Divide both sides by 144: x²/16 − y²/9 = 1.
This is the standard form x²/a² − y²/b² = 1 with a² = 16 and b² = 9, so a = 4 and b = 3.
Since there is a minus sign between the terms and both denominators are positive, this is a hyperbola with transverse axis along the x-axis.
Asymptotes: y = ±(b/a)x = ±(3/4)x.
Common mistakes
- Confusing c² = a² − b² (ellipse) with c² = a² + b² (hyperbola) → remember, ellipse has smaller c because e < 1.
- Using diameter instead of radius when writing a circle's equation → always check that you square the radius, not the diameter.
- Forgetting that for an ellipse a is always the larger denominator, so first rewrite the equation to identify a and b correctly.
- Plotting a parabola y² = 4ax as if it opens upward → it opens along the positive x-axis; x² = 4ay opens upward.
- Mixing up latus rectum formulas: 4a for parabola, 2b²/a for ellipse and hyperbola → keep a short list handy until they become automatic.
Quick revision
- Circle: all points equidistant from centre; e = 0.
- Parabola: e = 1; vertex, focus and directrix determine the curve; latus rectum = 4a.
- Ellipse: 0 < e < 1; sum of focal distances = 2a; c² = a² − b².
- Hyperbola: e > 1; difference of focal distances = 2a; c² = a² + b²; has two asymptotes.
- Standard equations assume centre (or vertex) at origin and axes along coordinate axes.