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JEE Conic Sections: Parabola

Parabola is the first conic section you’ll encounter when you start studying the conic sections. It is great for developing foundations of the conic sections. Once you’re well-versed with this topic, you won’t be facing much difficulties while studying ellipse and hyperbola.

Let’s start by picturing two lines intersecting each other. Now make one of the lines (called the Generating Line) revolve around the other. We get what is called a double cone. Let another plane now intersect it (obviously, not where the cone’s vertex lies–in which case you get straight lines!!).

What you get on the plane is called a conic section. The following are five conic sections and they are characterized by what is known as eccentricity (often denoted by ‘e’).

jee conic sections parabola, eccentricity - magooshSo let’s get started!

Basic Definition

We are given a straight line (directrix) and a point (focus). Then on a plane, the parabola is the set of all points such that they are equidistant from the directrix and the focus. The vertex is the point closest to the directrix. A line parallel to the directrix and passing through the focus cuts a conic section in two points, right? This particular line segment is called the latus rectum.

jee conic sections parabola, latus rectum - magoosh

By Melikamp – Own work, CC BY-SA 3.0, Link

Taking the directrix as x + a = 0 and the focus as (a, 0), the equation of a right open horizontal parabola is: y2 = 4 a x

jee conic sections parabola, right open horizontal parabola - magoosh

Thus, the parametric coordinates are: (at2, 2at)

For latus rectum, we set x = a and get: l = 4 a

Chords, Tangents, and Normals

jee conic sections parabola, chords - magoosh

It can be easily shown that a point (x1, y1) is an interior point if S1 > 0, lies on the parabola if S1 = 0 and exterior to it if S 1 < 0.

Focal Chord:
Any chord of the focus of a parabola is called a focal chord. It can be easily proved that if t1 and t2 are the parameters for its endpoints, then t1 t2 = –1.

Tangent

jee conic sections parabola, tangent - magoosh

jee conic sections parabola, tangent 2 - magoosh
(Complete description using Desmos Graphing Calculator)

The point of intersection of two tangents (drawn at t1 and t2) is (at1t2, a(t1+t2)).

Now, I am about to tell you something interesting! Just try to think over it. The tangents that are drawn at the end of a focal chord intersect at right angles on the directrix (Hint: t1t2 = –1 and hence m1m2 = –1). This also means that a circle drawn with a focal chord as the diameter touches the directrix! Don’t worry, see a complete proof here.

For any exterior point (x1, y1), the equation T2 = SS1 gives the tangents to the parabola from the given point. When S1 = 0, the point lies on the parabola and T = 0 which is actually true. Also notice that if S1 < 0, then T2 >0, and hence no tangent exists from the interior of a parabola.

When the tangents from an exterior point (x1, y1) are drawn, the points where they touch can be joined to form ‘The Chord of Contact’: T1 = 0

When an interior point (x1,y1) is given, then the chord through the point such it is the midpoint is given by:
T = S1

Don’t worry. The following examples will clear all your doubts.

Question 1
For the parabola y2 = 16 x, determine the angle between the tangents drawn from (–8,4).

    a) 37°
    b) 30°
    c) 72°
    d) 90°

Answer: c) 72°

Solution:
For the parabola y2 = 4 (4) x, a = 4
Putting (x1, y1) = (–8, 4) in the equation T2 = SS1 gives:
(4y — 2(4)(x — 8))2 = (y2 — 16x)(42 — 16(–8))
=> x2 — xy –2y2 + 20x + 8y + 64 = 0
=> (x + y + 4)(x — 2y + 16) = 0
Hence, the slopes are m1 = ½ and m2 = –1
The angle between the tangents is tanθ = mod((m2 — m1)/(1 — m1m2)) = 3
Since tanθ > 1 => θ > 45o
Also, θ < 90o
Checking the options, the option (c) is the most appropriate.

You should try more and more problems on this topic. After all, practice makes a man perfect!

Normal

Moving on, let us study the normal to a parabola!
jee conic sections parabola, normal - magoosh

jee conic sections parabola, normal 2 - magoosh
(Visualizing normals using Desmos Graphing Calculator)

Thus, if the slope of a normal is given, the equation of the normal can be rewritten as:

y = m x — (a m3 + 2 a m)

Moreover, you can even find the point to which the normal corresponds if its slope is given (using the fact that t = –m).

(at2, 2at) = (am2, –2am)

If you are given a point (x1, y1), then you can also find the possible slopes for a normal at that point.

y = m x — (am3 + 2am) ⇒ am3 + m (2a — x) + y = 0

Interesting! Through a single point on a parabola, as many as three normal could pass!

jee conic sections parabola, normal 3 - magoosh

Reflection property of a parabola: A ray of light parallel to the axis of parabola reflects and passes through the focus. This reflection property is exploited to make spherical mirrors used in physics to remove any spherical aberrations.

jee conic sections parabola, reflection property - magoosh

Image by Wikipedia

In this blog post, you learnt about some important concepts about Parabola. 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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