Circles and conic sections

NUST NET Maths · Maths · Analytic geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Each of these curves has a standard equation, and the test asks you to read its features straight from that equation. So the first step is always to put the equation in standard form, completing the square if x or y terms are present.

For a circle you read the centre and radius. For a parabola you find a from 4a, then the focus is a units inside the curve from the vertex and the directrix is a units outside. For an ellipse or hyperbola you find c, the distance from the centre to each focus, and the eccentricity e = c/a.

The eccentricity names the curve: 0 for a circle, less than 1 for an ellipse, exactly 1 for a parabola, more than 1 for a hyperbola.

Rules to remember

Circle (x − h)² + (y − k)² = r²: centre (h, k), radius r
Circle x² + y² + 2gx + 2fy + c = 0: centre (−g, −f), radius √(g² + f² − c); make the coefficients of x² and y² equal to 1 first
Tangent to x² + y² = r² at (x₁, y₁): xx₁ + yy₁ = r²; the line y = mx + c touches this circle when c² = r²(1 + m²)
Parabola y² = 4ax: vertex (0, 0), focus (a, 0), directrix x = −a. Parabola x² = 4ay: focus (0, a), directrix y = −a
Ellipse x²/a² + y²/b² = 1 with a > b: c² = a² − b², e = c/a, foci (±c, 0), directrices x = ±a²/c
If the larger number is under y², the major axis is along the y-axis and the foci are (0, ±c)
Hyperbola x²/a² − y²/b² = 1: c² = a² + b², e = c/a, foci (±c, 0), directrices x = ±a²/c

Common mistake

Mixing up the two formulas for c: the ellipse uses c² = a² − b² but the hyperbola uses c² = a² + b². Check your answer against e < 1 for an ellipse and e > 1 for a hyperbola.

Worked example

Find the focus of the parabola x² − 4x − 8y + 12 = 0.

Complete the square: (x − 2)² − 4 = 8y − 12, so (x − 2)² = 8(y − 1).
Compare with X² = 4aY: vertex (2, 1) and 4a = 8, so a = 2.
The parabola opens upwards, so the focus is 2 units above the vertex: (2, 1 + 2).

Answer: Focus (2, 3); the directrix is y = −1.

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Coordinate Geometry

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