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JEE Conic Sections: Ellipse

The topic ‘conic sections’ comprises of parabola, ellipse, and hyperbola. This is one of the most important topics of JEE Mathematics section, for both JEE Mains as well as JEE Advanced. You can expect two to five questions from conic sections in JEE Mains and four to seven questions in JEE Advanced. Such questions may involve one or more than one conic sections, so you need to be well prepared in all three of them. All in all, this is an extremely important topic and demands full attention and adequate practice from your side.

So, let’s learn about ellipses in this blog post!

What Is an Ellipse?

Hey there! Let’s start this chapter with a small exercise. No, not the JEE problems, but a simple, fun exercise. Pick up any coin from your wallet. Found it? Now switch off the lights and shine a torch or flashlight onto the coin pressed against a wall. Slowly pull the coin away from the wall. You’d see a shadow of the circle. Now, rotate the coin slightly in any direction. What shaped shadow did you observe? It looks like a squashed circle, doesn’t it? Well, we have a word for it–ellipse!

Ellipse As a Conic Section

Ellipse is a part of conic sections, that is, it can be obtained as a cross-section of a cone. If you cut a cone parallel to its base, you get a circle. But if you make a cut on the cone at an angle through its curved face, you get an ellipse, as shown in the figure below. More precisely, an ellipse is any plane section of a cone, not containing the cone’s apex, and with a slope less than the slope of the lines on the cone.
In fact, you could call a circle a special type of ellipse!

jee conic sections ellipse, cross section of cone - magoosh

By Ag2gaeh – अपना काम, CC BY-SA 4.0, Link

Mathematical Description of an Ellipse

Now that we have a brief idea of what an ellipse is and what it looks like, let’s describe the ellipse mathematically.

Ellipse can be described as a curve in which the ratio of the distance between any point P on the curve and a fixed point F, and the distance between that point (P) and a fixed line L is a constant real number greater than/equal to zero and less than one.

Too much jargon? Don’t worry. Let’s break it down step-by-step.

Take a fixed point F, say (0, 0).

jee conic sections ellipse, mathematical ellipse graph 1 - magoosh

Now take a fixed line L, say x = 1, or, x — 1 = 0.

jee conic sections ellipse, mathematical ellipse graph 2 - magoosh

Now, take a general point P(x, y). Drop a perpendicular from point P onto line L. Let the perpendicular from P and line L intersect at P’, that is, the (perpendicular) distance between P and L is PP’. The distance between P and F is PF.

We have,

PF/PP’ = constant ∊ (0, 1). Let’s set this constant to be 0.5.
⇒PF/PP’ = 0.5
⇒√((x — 0)2 + (y — 0)2)/(|x — 1|/√(12)) = 0.5
⇒2√(x2 + y2) = |x — 1|
⇒4(x2 + y2) = (x — 1)2
⇒4x2 + 4y2 = x2–2x + 1
⇒3x2 + 4y2 + 2x — 1 = 0 is the equation of the ellipse.

If you sketch the curve, you’d get something like this:

jee conic sections ellipse, mathematical ellipse graph 3 - magoosh

The curve in green is the desired ellipse. Every point on this curve is twice as far from the line x — 1 = 0 as it is from the point (0, 0). You can check and verify it for any point on this ellipse.
The fixed point F is known as ‘focus’ of the ellipse, and the fixed line L is known as the ‘directrix’ of the ellipse.

The ratio, (Distance of any point from Focus)/(Distance of the same point from Directrix) is known as the eccentricity of the ellipse, denoted by ‘e’, where 0 < = e < 1 Eccentricity is a measure of the degree of resemblance of an ellipse to a circle. For a circle, eccentricity will be zero. That is why we consider a circle as a special type of ellipse.

Every ellipse has 2 foci and 2 corresponding directrices, due to symmetry. Given a focus and a directrix, we can uniquely define an ellipse. Likewise, an ellipse can be uniquely defined by giving the focus and its eccentricity.

Another Description

Ellipse can also be described in another, simpler way. Ellipse can be described as a curve in which the sum of distances between any point P on the curve and 2 fixed points is a constant number, greater than the distance between the fixed points.

Got confused? No worries, let’s simplify this as well.

Take 2 points, say F1(–4, 0) and F2(4, 0). Now, take a general point P(x, y). The distance of the point P from F1 is given by PF1 = √((x– (–4))2 +(y — 0)2). Similarly, the distance of the point P from F2 is given by PF2 = √((x — 4)2 + (y — 0)2). The distance between F1 and F2 is √((4– (–4))2 + (0 — 0)2) = 8 units.

We have, PF1 + PF2 = constant, where constant > F1F2
⇒√((x– (–4))2 + (y-0)2) + √((x — 4)2 + (y — 0)2) = constant, constant > 8

Let’s take constant = 10

⇒√(x2 + 8x + 16 + y2) + √(x2 –8x + 16 + y2) = 10
⇒√(x2 + 8x + 16 + y2) = 10 — √(x2 — 8x + 16 + y2)
⇒x2 + 8x + 16 + y2 = 100 + x2 — 8x + 16 + y2 — 20√(x2 –8x + 16 + y2)
⇒25 — 4x = 5√(x2 –8x + 16 + y2)
⇒625 + 16x2 — 200x = 25x2 — 200x + 400 + 25y2
⇒9x2 + 25y2 — 225 = 0 is the desired ellipse

jee conic sections ellipse, ellipse as a curve - magoosh

If you take any point P(x, y) located on this ellipse, the sum of its distances from the points F1(–4, 0) and F2(4, 0) will be 10. Try to check and confirm it!

In general, for any ellipse, the fixed points F1 and F2 are known as the ‘foci’ of the ellipse. Yes, these are the same foci that we discussed in the previous description of foci and directrices. Isn’t it interesting? Hold your excitement, there is a lot more interesting stuff coming up.

Also, the fixed sum of distances is known as the length of ‘major axis’ of the ellipse, denoted by ‘2a’. The major axis is the line joining the two farthest points in a given ellipse. For example, in the ellipse described above, the x-axis is the major axis as it is the line joining (–5, 0) and (5, 0), which are clearly the farthest points in that ellipse. Length of the major axis is the distance between the farthest points, given by √((5–(–5))2 + (0 — 0)2) = 10 units.

Also, the ratio (F1F2)/(2a) is nothing but the eccentricity (e) of the given ellipse, that is,
e = (F1F2)/(2a) = 8/10 = 0.8

Note that this is the same eccentricity about which we learned in the previous section. It comes full circle.

Summary

Let’s quickly summarize our discussion on ellipse,

  • Ellipse is any plane section of a cone, not containing the cone’s apex, and with a slope less than the slope of the lines on the cone.
  • (Distance of any point on ellipse from a fixed point called Focus)/(Distance of the same point from a fixed point called Directrix) is known as eccentricity, denoted by ‘e’.The eccentricity of the ellipse lies in the range 0 < = e < 1.
  • Every ellipse has 2 foci and 2 corresponding directrices, due to symmetry.
  • An ellipse can be described as a curve in which the sum of distances between any point P on the curve and 2 fixed points called foci of the ellipse, is a constant number known as the length of ‘major axis’ of the ellipse. This is denoted by ‘2a’, which must be greater than the distance between the foci.
  • The ratio of the distance between the foci and the length of the major axis is the eccentricity (e) of the given ellipse.

In this post, we learned some important theoretical concepts of an Ellipse. Hope you found it useful. For more amazing JEE study material, check out Magoosh JEE Product. Happy learning!

For more information about JEE Conic Sections, check out the following Magoosh resources:

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