A satellite dish is parabolic because a parabola:

Reflects rays parallel to its axis through a single focus

For an ellipse, the sum of the distances from any point to the two foci equals:

2a, the length of the major axis

Conic Sections — Parabola, Ellipse and Hyperbola

The three conics look like three separate topics and are really one, because all of them are defined the same way: the locus of a point whose distance from a fixed point bears a constant ratio to its distance from a fixed line. That ratio is the eccentricity, and it alone decides which curve you get — less than one gives an ellipse, exactly one a parabola, more than one a hyperbola. Recognising that means every formula in the chapter comes in three parallel versions rather than eighteen unrelated ones, and the pattern is worth learning in preference to the individual results.

The Three Conics Side by Side

Standard forms and their parameters
Feature	Parabola y² = 4ax	Ellipse x²/a² + y²/b² = 1	Hyperbola x²/a² − y²/b² = 1
Eccentricity	e = 1	e < 1	e > 1
Relation among a, b, e	Not applicable	b² = a²(1 − e²)	b² = a²(e² − 1)
Focus	(a, 0)	(±ae, 0)	(±ae, 0)
Directrix	x = −a	x = ±a/e	x = ±a/e
Latus rectum	4a	2b²/a	2b²/a
Tangent condition for y = mx + c	c = a/m	c² = a²m² + b²	c² = a²m² − b²

🧠 The only sign that changes is the one in front of b²

Compare the ellipse and hyperbola columns and almost everything is identical: same focus form, same directrix, same latus rectum, same shape of tangent condition. The single difference is a sign, because b² = a²(1 − e²) for the ellipse and a²(e² − 1) for the hyperbola, and that propagates into c² = a²m² + b² versus a²m² − b². So instead of memorising two sets of results you can learn one set and remember that the hyperbola flips the sign in front of b². This also explains why an ellipse is a closed curve while a hyperbola opens out: with e < 1 the quantity 1 − e² is positive and bounds the curve, and with e > 1 it goes negative and the bound disappears.

Each conic has a focal definition that is often the fastest route to an answer. For an ellipse the sum of the distances to the two foci is constant and equals 2a; for a hyperbola the difference of those distances is constant and equals 2a; and for a parabola the distance to the focus equals the distance to the directrix. A question describing a locus in those words is handing you the conic and its parameter without any algebra at all.

Tangents, Normals and Special Properties

The tangent at a point on any conic is obtained by the same substitution rule T = 0: replace x² by xx₁, y² by yy₁, x by (x + x₁)/2 and y by (y + y₁)/2. So the tangent to y² = 4ax at (x₁, y₁) is yy₁ = 2a(x + x₁), and the tangent to the ellipse is xx₁/a² + yy₁/b² = 1. Using the parametric point is often cleaner still: (at², 2at) on the parabola, (a cos θ, b sin θ) on the ellipse and (a sec θ, b tan θ) on the hyperbola.

🎯 Why a satellite dish is a parabola and a whispering gallery is an ellipse

The reflective property of a parabola is that every ray arriving parallel to the axis is reflected through the focus, which is exactly what a dish antenna or a solar cooker needs — collect a wide beam and concentrate it at one point. Run it backwards and a source at the focus produces a parallel beam, which is what a headlight does. The ellipse reflects a ray from one focus to the other, so a sound made at one focus of an elliptical room converges audibly at the second focus even across a large hall. Both properties follow from the focal definitions rather than being extra facts, and questions that mention a dish, a mirror or a gallery are naming the conic for you.

Properties asked about directly
Property	Statement
Focal chord of a parabola	If one end is at parameter t, the other is at −1/t
Shortest focal chord	The latus rectum itself
Director circle of an ellipse	x² + y² = a² + b², the locus of perpendicular tangents
Asymptotes of x²/a² − y²/b² = 1	y = ±(b/a)x
Rectangular hyperbola	a = b, so e = √2 and the asymptotes are perpendicular
Conjugate hyperbola	y²/b² − x²/a² = 1, sharing the same asymptotes

⚠️ Identify which axis is major before using a or b

In x²/a² + y²/b² = 1 the convention is that a > b, so the major axis lies along x and the foci are at (±ae, 0). But an equation such as x²/9 + y²/25 = 1 has the larger denominator under y², so the major axis is vertical, the roles of a and b swap, and the foci are on the y-axis at (0, ±4). Applying the standard formulae without checking which denominator is larger puts the foci on the wrong axis and makes every subsequent answer wrong. The safe habit is to write down which axis is major as the first line of the solution, before any formula is used.

Key takeaways

All three conics share one definition; the eccentricity alone decides which curve you get.
e < 1 ellipse, e = 1 parabola, e > 1 hyperbola; a rectangular hyperbola has e = √2.
b² = a²(1 − e²) for an ellipse and a²(e² − 1) for a hyperbola — only the sign changes.
Focal definitions: sum of focal distances is 2a for an ellipse, the difference is 2a for a hyperbola.
Latus rectum is 4a for a parabola and 2b²/a for both the ellipse and hyperbola.
T = 0 gives the tangent for every conic; parametric points are often quicker still.
A parabola reflects parallel rays to its focus — that is what a dish antenna exploits.
Check which denominator is larger before locating the foci — the major axis may be vertical.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

The eccentricity of a rectangular hyperbola is:
1. 1
2. √2
3. 2
4. 1/√2
Show answer
Answer: B — √2
A rectangular hyperbola has a = b, and substituting into b² = a²(e² − 1) gives a² = a²(e² − 1), so e² = 2 and e = √2. The name comes from the asymptotes y = ±(b/a)x becoming y = ±x, which are perpendicular. An eccentricity of 1 would be a parabola, so that option can be eliminated on sight.
For the ellipse x²/9 + y²/25 = 1, the foci lie:
1. On the x-axis at (±4, 0)
2. On the y-axis at (0, ±4)
3. On the x-axis at (±5, 0)
4. At the origin
Show answer
Answer: B — On the y-axis at (0, ±4)
The larger denominator, 25, sits under y², so the major axis is vertical and the roles of a and b are interchanged: a = 5 along y and b = 3 along x, giving c = √(25 − 9) = 4 and foci at (0, ±4). Applying the default formula and putting the foci on the x-axis is the standard error, and it is why checking which denominator is larger must come first.
The length of the latus rectum of y² = 12x is:
1. 3
2. 6
3. 12
4. 24
Show answer
Answer: C — 12
Comparing with y² = 4ax gives 4a = 12 so a = 3, and the latus rectum of a parabola is 4a, which is 12 — the same as the coefficient of x, a shortcut worth noticing. The answer 3 is the value of a itself and 6 is the semi-latus rectum, both of which are one step short of the quantity asked for.
A satellite dish is parabolic because a parabola:
1. Has two foci
2. Reflects rays parallel to its axis through a single focus
3. Is a closed curve
4. Has perpendicular asymptotes
Show answer
Answer: B — Reflects rays parallel to its axis through a single focus
The reflective property concentrates an incoming parallel beam at the focus, which is what lets a dish collect a weak signal over a large area and deliver it to a single receiver; run in reverse it produces the parallel beam of a headlight. A parabola has only one focus and is an open curve, and asymptotes belong to the hyperbola — so each of the other options describes a different conic.
For an ellipse, the sum of the distances from any point to the two foci equals:
1. 2a, the length of the major axis
2. 2b
3. 2ae
4. It varies from point to point
Show answer
Answer: A — 2a, the length of the major axis
The constancy of that sum is the focal definition of an ellipse and equals the major axis length 2a, which is why the curve can be drawn with a loop of string round two pins. For a hyperbola it is the difference of the two distances that stays constant at 2a — the same number, a different operation, and that contrast is a frequent one-mark question.
The asymptotes of x²/16 − y²/9 = 1 are:
1. y = ±x
2. y = ±(3/4)x
3. y = ±(4/3)x
4. x = ±4
Show answer
Answer: B — y = ±(3/4)x
For x²/a² − y²/b² = 1 the asymptotes are y = ±(b/a)x, and here a = 4 and b = 3, giving slopes ±3/4. Inverting the ratio to 4/3 is the usual slip. The asymptotes would be y = ±x only for a rectangular hyperbola with a = b, and they are always lines through the centre rather than vertical lines.
If one end of a focal chord of y² = 4ax has parameter t, the other end has parameter:
1. t
2. −t
3. 1/t
4. −1/t
Show answer
Answer: D — −1/t
For the parametric points (at₁², 2at₁) and (at₂², 2at₂) to be collinear with the focus (a, 0), the parameters must satisfy t₁t₂ = −1, so the second is −1/t. This immediately gives the shortest focal chord: it occurs when t = 1 and t = −1, that is at the latus rectum, whose length 4a is therefore the minimum.
The condition for y = mx + c to be tangent to the ellipse x²/a² + y²/b² = 1 is:
1. c = a/m
2. c² = a²m² + b²
3. c² = a²m² − b²
4. c = 0
Show answer
Answer: B — c² = a²m² + b²
Substituting the line into the ellipse and setting the discriminant to zero gives c² = a²m² + b². The hyperbola condition is the same expression with a minus sign, and the parabola condition c = a/m has a different form entirely — so the three are best remembered as one pattern with the sign of b² as the only variable, rather than as three unrelated results.