Ellipse — Introduction

Take two pins, push them into a board about 10 cm apart, and loop a piece of string around both pins. Pull the string taut with a pencil and trace the curve, keeping the string taut at all times. The shape you draw is an ellipse.

Why does this work? At every position of the pencil, the total length of string is fixed. That total length is split into two pieces — one from the pencil to the left pin, one from the pencil to the right pin. So the pencil traces all points where the sum of the distances to the two pins is a constant. That is the definition of an ellipse.

The two pins are called the foci (singular: focus). The string length minus the distance between the pins determines how fat the ellipse is. If the string is much longer than the pin spacing, the ellipse is nearly circular. If the string is barely longer, the ellipse is thin and elongated. The extreme case — string length equals pin spacing — collapses the ellipse to the line segment between the pins. The other extreme — pins on top of each other — gives a perfect circle.

This is not just a classroom construction. The orbits of planets around the Sun are ellipses, with the Sun at one focus. Kepler discovered this in 1609, using astronomical data from Tycho Brahe, and it overturned two thousand years of circular-orbit astronomy. Indian astronomers, including Aryabhata and Brahmagupta, had developed sophisticated models of planetary motion centuries earlier, and the elliptical framework that Kepler formalised ultimately provided the geometric explanation for the deviations they had observed.

The definition, precisely

The notation is worth unpacking.

• a is a positive number. The constant sum of distances is 2a.
• c is the distance from the centre (midpoint of F_1F_2) to each focus. So F_1 and F_2 are at (\pm c, 0) when the ellipse is centred at the origin with foci on the x-axis.
• The constraint 2a > 2c (equivalently, a > c) is necessary. If a = c, the "ellipse" degenerates to a line segment. If a < c, no point satisfies the condition at all — the string is too short to reach both pins.

There is also a focus-directrix definition, analogous to the parabola's: an ellipse is the locus of a point whose distance from a fixed point (focus) divided by its distance from a fixed line (directrix) equals a constant e < 1. The two definitions are equivalent. The focus-directrix form is useful in some JEE problems, but the "sum of distances" form is the one that builds intuition, so we start here.

Deriving the standard equation

Place the centre of the ellipse at the origin. Put the foci on the x-axis at F_1(-c, 0) and F_2(c, 0). Take any point P(x, y) on the ellipse. The defining condition is

Write out the distances explicitly:

This looks messy, but it simplifies beautifully. The key is to isolate one radical and square.

Step 1. Isolate one radical.

Why: move the second radical to the right so you can square both sides and eliminate one square root at a time.

Step 2. Square both sides.

Expand (x + c)^2 = x^2 + 2cx + c^2 and (x - c)^2 = x^2 - 2cx + c^2. The x^2, y^2, and c^2 terms appear on both sides and cancel:

Divide by -4:

Why: after expanding, most terms cancel in pairs. The remaining radical is now isolated on one side — ready for a second squaring.

Step 3. Square again.

Expand the left side:

The -2a^2cx terms cancel from both sides:

Rearrange:

Why: collect the x^2 terms and factor. The expression a^2 - c^2 appears on both sides — and since a > c, this quantity is positive.

Step 4. Define b^2 = a^2 - c^2.

Since a > c > 0, we have b > 0. Substitute:

Divide both sides by a^2 b^2:

Why: dividing by a^2 b^2 normalises each term to 1, giving the cleanest possible form. The constant on the right is 1 — always. This is the standard form of an ellipse.

Notice that the derivation never assumed anything about where P is — it works for every point on the curve. And the relation b^2 = a^2 - c^2 (equivalently, c^2 = a^2 - b^2) connects the three key parameters. Given any two, you can find the third.

The anatomy of an ellipse

Every term has a geometric meaning. Here they all are, in one picture.

• Centre (C): The midpoint of the segment joining the two foci. In the standard form, it is the origin (0, 0).

• Foci (F_1, F_2): The two fixed points. Located at (-c, 0) and (c, 0), where c = \sqrt{a^2 - b^2}.

• Major axis: The longest diameter of the ellipse, passing through both foci. Its length is 2a. The endpoints are called the vertices A(a, 0) and A'(-a, 0).

• Minor axis: The shortest diameter, perpendicular to the major axis through the centre. Its length is 2b. The endpoints are B(0, b) and B'(0, -b).

• Semi-major axis (a): Half the length of the major axis.

• Semi-minor axis (b): Half the length of the minor axis.

The relationship between a, b, and c is the Pythagorean-like identity a^2 = b^2 + c^2. If you think of a right triangle with hypotenuse a, one leg b, and the other leg c, the geometry of the ellipse is encoded in that triangle. One way to see this: put a point at B(0, b) on the ellipse. Its distances to the foci are \sqrt{c^2 + b^2} = \sqrt{c^2 + a^2 - c^2} = a. So BF_1 + BF_2 = 2a, confirming the definition. The triangle F_1 B C has legs c and b and hypotenuse a.

Eccentricity

The eccentricity of an ellipse is the ratio

It measures how "un-circular" the ellipse is.

• If c = 0 (the two foci coincide at the centre), then e = 0 and b = a, so the equation becomes x^2/a^2 + y^2/a^2 = 1, which is x^2 + y^2 = a^2 — a circle. A circle is an ellipse with eccentricity zero.

• If c is close to a (the foci are near the vertices), then e is close to 1 and b is close to 0. The ellipse is very thin and elongated.

• The eccentricity always satisfies 0 \leq e < 1 for an ellipse. At e = 1, the curve is a parabola (the "second focus" has moved to infinity). At e > 1, the curve is a hyperbola.

Since c^2 = a^2 - b^2, you can also write e = \sqrt{1 - b^2/a^2}, or equivalently, b^2 = a^2(1 - e^2).

The Earth's orbit around the Sun is an ellipse with e \approx 0.0167 — almost a perfect circle, but not quite. The orbit of Halley's Comet has e \approx 0.967 — so elongated that it spends most of its orbit far from the Sun.

The focus-directrix definition

There is a second way to define an ellipse, analogous to how the parabola is defined.

For the standard ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, each focus has a corresponding directrix:

• Focus F_1(-c, 0), directrix x = -a/e = -a^2/c
• Focus F_2(c, 0), directrix x = a/e = a^2/c

The directrix lies outside the ellipse (since a/e = a^2/c > a). The ratio PF/Pd = e is always less than 1, meaning the point is always closer to the focus than to the directrix.

Verification. Take a point (x, y) on the ellipse. Its distance from F_2(c, 0) is \sqrt{(x-c)^2 + y^2}. From the derivation of the standard form, you saw that a\sqrt{(x-c)^2 + y^2} = a^2 - cx, so the distance is (a^2 - cx)/a = a - ex. The distance from the directrix x = a/e is a/e - x. The ratio is

This confirms the equivalence.

Worked examples

Common confusions

• "The larger number is always a." Yes — by convention, a > b for an ellipse. If you see \frac{x^2}{4} + \frac{y^2}{9} = 1, then a^2 = 9 (not 4) and the major axis is along the y-axis. The larger denominator identifies a^2, regardless of which variable it sits under.

• "b is the distance from the centre to a focus." No. c is the distance from the centre to a focus. b is the semi-minor axis — the distance from the centre to the top (or bottom) of the ellipse. They are related by b^2 = a^2 - c^2, but they are not the same. In fact, b < a and c < a, but b and c have no fixed ordering.

• "Eccentricity can be 1." Not for an ellipse. If e = 1, then c = a and b = 0, and the curve degenerates. A conic with e = 1 is a parabola; a conic with e < 1 is an ellipse; a conic with e > 1 is a hyperbola.

• "An ellipse is just a stretched circle." This is actually true — and it is a useful way to think about it. If you take the circle x^2 + y^2 = a^2 and compress the y-coordinate by the factor b/a, every point (x, y) maps to (x, by/a), and the circle maps to the ellipse x^2/a^2 + y^2/b^2 = 1. This scaling is the basis of the auxiliary circle construction, which you will see in the next article.

• "The sum of distances is a." The sum is 2a, not a. The semi-major axis is a; the full constant sum is twice that. Forgetting the factor of 2 is one of the most common arithmetic mistakes in ellipse problems.

Going deeper

The ellipse as a conic section

The word "conic" comes from cone. If you take a right circular cone (like an ice-cream cone extended infinitely in both directions) and slice it with a plane, the cross-section is a conic section. The type of conic depends on the angle of the slice:

• Slice perpendicular to the cone's axis: circle.
• Slice at a moderate angle (cutting through one nappe of the cone): ellipse.
• Slice parallel to one slant edge of the cone: parabola.
• Slice at a steep angle (cutting through both nappes): hyperbola.

The ellipse is the "in-between" case — less special than a circle (which requires an exact perpendicular cut) but more bounded than a parabola or hyperbola (which extend to infinity).

The equation with foci on the y-axis

If the major axis is vertical instead of horizontal, the foci are at (0, \pm c) and the equation is

with a > b > 0 and c = \sqrt{a^2 - b^2}. Everything works the same way — vertices at (0, \pm a), co-vertices at (\pm b, 0), directrices at y = \pm a/e. The rule is simple: the variable under the larger denominator identifies the direction of the major axis.

The general ellipse

The standard form places the centre at the origin and the axes along the coordinate axes. A general ellipse, shifted to centre (h, k) but with axes still parallel to the coordinate axes, has the equation

This is just a translation. The foci move to (h \pm c, k), the vertices to (h \pm a, k), and so on.

An even more general ellipse — one whose axes are tilted at an angle to the coordinate axes — has the equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, subject to B^2 - 4AC < 0 (and A \neq C or B \neq 0, to exclude the circle). Handling the tilt requires rotation of axes, which is a separate topic.

The reflective property (preview)

Just as a parabola reflects parallel rays through its focus, an ellipse reflects any ray starting from one focus to the other focus. If you stand at one focus of an elliptical room and whisper, the sound waves bounce off the walls and converge at the other focus. This is the principle behind whispering galleries — and the Gol Gumbaz at Bijapur (Vijayapura) in Karnataka, one of the world's most famous acoustic structures, demonstrates this effect spectacularly, though its geometry is a dome rather than an ellipse. The precise reflective property of the ellipse will be derived in a later article.

A Pythagorean identity in disguise

The relation a^2 = b^2 + c^2 looks exactly like the Pythagorean theorem — and it is. Consider the right triangle formed by the centre C(0,0), a focus F(c, 0), and the co-vertex B(0, b). The segment CF has length c (one leg), the segment CB has length b (the other leg), and the segment BF — the distance from the co-vertex to the focus — has length \sqrt{c^2 + b^2} = \sqrt{a^2} = a (the hypotenuse). So the triangle CBF has legs b and c and hypotenuse a. The semi-major axis is the hypotenuse of the right triangle formed by the semi-minor axis and the focal distance.

This is not a coincidence. It is the geometric content of the definition: at the co-vertex, the two focal distances are equal (both equal to a, since the sum is 2a and symmetry forces them to be the same), and each focal distance is the hypotenuse of a right triangle with legs b and c.

Where this leads next

The ellipse is the second conic you have met, after the parabola. The next articles build on this foundation.

• Ellipse — Auxiliary Circle and Eccentric Angle — the parametric form (a\cos\theta, b\sin\theta), the auxiliary circle, the director circle, and the latus rectum.
• Ellipse — Tangent and Normal — equations of tangent and normal in all forms, the condition for tangency, and the reflective property.
• Hyperbola — Introduction — the next conic section, where the difference of focal distances is constant instead of the sum.
• Parabola — Introduction — the limiting case e = 1, if you want to revisit where this story began.
• Locus — the general technique for translating geometric conditions into algebraic equations, used throughout this article.