Comparison of conics and applications

The four conics , circle, parabola, ellipse, hyperbola , share many features and differ in a few crucial ones. Mastery of this chapter means being able to tell them apart at a glance and recall the relevant formulas instantly.

Side-by-side summary

Feature	Circle	Parabola	Ellipse	Hyperbola
Standard form	$x^2 + y^2 = r^2$	$y^2 = 4 a x$	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$	$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$
Eccentricity	$0$	$1$	$0 < e < 1$	$e > 1$
Foci	centre only	$(a, 0)$	$(\pm c, 0)$	$(\pm c, 0)$
$c^2$ relation	$-$	$-$	$a^2 - b^2$	$a^2 + b^2$
Sum / diff of focal distances	$-$	$-$	sum $= 2 a$	$\\|\text{diff}\\| = 2 a$
Latus rectum length	$2 r$ (diameter)	$4 a$	$\dfrac{2 b^2}{a}$	$\dfrac{2 b^2}{a}$
Asymptotes	none	none	none	$y = \pm \dfrac{b}{a} x$
Directrix	none (degenerate)	$x = -a$	$x = \pm a/e$	$x = \pm a/e$

How to tell which is which

Suppose you are given a general second-degree equation in $x, y$ . Reduce to a "standard"-looking form by completing the square and dividing.

Equal positive coefficients on $x^2$ and $y^2$ , no $xy$ : circle.
One quadratic term only: parabola.
Both quadratic terms positive, unequal: ellipse.
Quadratic terms with opposite signs: hyperbola.

Applications

Projectile motion. A particle launched with initial velocity $v$ at angle $\theta$ above the horizontal traces a parabola. The horizontal range and maximum height come from the standard parabola equations.

Planetary orbits. Each planet orbits the Sun in an ellipse with the Sun at one focus. The eccentricity of Earth's orbit is about $0.017$ , very nearly circular.

Reflective properties.

Parabola. Rays parallel to the axis reflect through the focus. (Satellite dishes; car headlights working in reverse.)
Ellipse. A ray emitted from one focus reflects through the other focus.
Hyperbola. A ray heading toward one focus, when it strikes the near branch, reflects so as to seem to come from the other focus.

Cooling towers, headlights, telescopes. All conics make engineering appearances; the most common are paraboloid telescopes and elliptical mirrors.

Worked examples

Example 1. Identify and find key parameters: $4 x^2 + 9 y^2 - 16 x + 18 y - 11 = 0$ .

Complete squares: $4(x^2 - 4x) + 9(y^2 + 2y) = 11 \Rightarrow 4(x - 2)^2 + 9(y + 1)^2 = 11 + 16 + 9 = 36 \Rightarrow \dfrac{(x - 2)^2}{9} + \dfrac{(y + 1)^2}{4} = 1$ . Ellipse, centre $(2, -1)$ , $a = 3, b = 2$ , $c = \sqrt{5}$ .

Example 2. Identify and find key parameters: $9 y^2 - 4 x^2 - 18 y - 8 x - 31 = 0$ .

Complete squares: $9(y^2 - 2 y) - 4(x^2 + 2 x) = 31 \Rightarrow 9(y - 1)^2 - 4(x + 1)^2 = 31 + 9 - 4 = 36 \Rightarrow \dfrac{(y - 1)^2}{4} - \dfrac{(x + 1)^2}{9} = 1$ . Hyperbola, centre $(-1, 1)$ , transverse axis vertical, $a = 2, b = 3$ .

Example 3. Find the equation of the parabola whose vertex is at $(0, 0)$ , axis along the $x$ -axis, and which passes through $(4, -8)$ .

$y^2 = 4 a x \Rightarrow 64 = 16 a \Rightarrow a = 4$ . Equation: $y^2 = 16 x$ .

Example 4. A ladder of length $10$ slides down a wall, keeping its top on the wall and bottom on the floor. Find the locus of its midpoint.

Let the ladder reach $(0, b)$ on the wall and $(a, 0)$ on the floor, with $a^2 + b^2 = 100$ . Midpoint: $(a/2, b/2)$ . So $(2 x)^2 + (2 y)^2 = 100 \Rightarrow x^2 + y^2 = 25$ . A circle of radius $5$ .

Example 5. Find the eccentricity of $\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1$ .

Vertical major. $a = 4, b = 5$ . $c^2 = b^2 - a^2 = 9 \Rightarrow c = 3$ . $e = c/b = 3/5$ .

Try it yourself

Identify $x^2 + 4 y^2 - 2 x + 16 y - 11 = 0$ .
Identify $2 x^2 - 3 y^2 + 8 x - 6 y - 11 = 0$ .
Sketch and identify $x^2 = -12 y$ .
Find eccentricity of $\dfrac{x^2}{49} + \dfrac{y^2}{36} = 1$ .
A point moves so its distance from $(2, 3)$ equals its distance from line $y = -1$ . Find and identify.
Find the foci of $\dfrac{(x - 1)^2}{16} + \dfrac{(y + 2)^2}{9} = 1$ .
Find the asymptotes of $\dfrac{(x - 3)^2}{4} - \dfrac{(y + 1)^2}{9} = 1$ .
Find the equation of the locus of a point $P$ such that $|PA| + |PB| = 10$ where $A = (-3, 0), B = (3, 0)$ .
A man on a tower sees a satellite tracing a circular orbit. The tower's shadow at a fixed time has length $h$ . Is the locus of shadow tips a conic? Identify.
Find the locus of the midpoints of chords of the circle $x^2 + y^2 = 25$ that pass through the point $(1, 2)$ .
A point moves so its distance from the origin is half its distance from the line $x = 4$ . Identify the locus.
Find the eccentricity, foci, and equation of directrices of $4 x^2 + 9 y^2 - 36 = 0$ .

Pitfalls / Tricks

Always complete the square to identify a shifted conic.
The sign before $y^2$ is the most important clue: $+$ with $x^2$ means ellipse/circle; $-$ means hyperbola.
A degenerate conic (like $x^2 - y^2 = 0$ ) factors into two lines , be alert.
Insight. The four conics form a continuous family parametrised by eccentricity. As $e$ slides from $0$ to $\infty$ : circle → ellipse → parabola → hyperbola.