The Parabola

Name: The Parabola
Author: StemPy Lab

Throw a ball and it traces a parabola.

Hold a torch at the focus of a mirror and the beam goes out straight.
This course defines the parabola as the set of points as far from a fixed point as from a fixed line.
You get the four standard equations, the latus rectum, the parametric form, the tangent and the normal in three forms, the focal chord and the reflection property.
It covers the parabola in the Indian syllabus unit (NCERT, JEE Main, BITSAT) and in A-Level Further Maths.

3 lessons, about 1 h 30 min3 simulations10 problems3 practice questionsPython in every lesson

Course outline (3 lessons)

Lesson 1about 30 min
The parabola from focus and directrix
A parabola is the set of points that are as far from a fixed point, the focus, as from a fixed line, the directrix. With the focus $(a, 0)$ and the directrix $x = -a$ this gives $y^2 = 4ax$ . There are four standard forms, and the latus rectum has length $4a$ . Every point is $(at^2, 2at)$ for one value of $t$ .
1 simulation1 practice question1 Python program4 problems1 Python lab
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Lesson 2about 30 min
Tangents and normals to a parabola
The tangent at $(at^2, 2at)$ is $ty = x + at^2$ and the normal is $y + tx = 2at + at^3$ . In terms of the slope $m$ the tangent is $y = mx + \dfrac{a}{m}$ and the normal is $y = mx - 2am - am^3$ . From a point outside, two tangents can be drawn and up to three normals. Perpendicular tangents meet on the directrix.
1 simulation1 practice question1 Python program4 problems1 Python lab
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Lesson 3about 30 min
Focal chords and the reflection property
A chord through the focus has its ends at parameters $t$ and $-\dfrac{1}{t}$ . Its length is $a\left(t + \dfrac{1}{t}\right)^2$ , and the two parts of it satisfy $\dfrac{1}{SP} + \dfrac{1}{SQ} = \dfrac{1}{a}$ . The tangents at its ends are perpendicular and meet on the directrix. The tangent also makes equal angles with the focal line and the axis, which is why a dish gathers parallel rays at the focus.
1 simulation1 practice question1 Python program2 problems1 question set1 note
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JEE Main format20 questions · 50 min

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