The Ellipse and the Hyperbola

Name: The Ellipse and the Hyperbola
Author: StemPy Lab

A planet circles the Sun on an ellipse, and a sonic boom spreads along a hyperbola.

Both curves come from the same focus and directrix idea as the parabola, with the ratio e a little different.
You write the standard equations, find the foci, the latus rectum and the eccentricity, and use the auxiliary circle, the parametric form, tangents, normals and the asymptotes.
You end by telling a conic from its general equation.
It covers the second half of the conics unit in the Indian syllabus (NCERT, JEE Main) 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 ellipse
An ellipse is the set of points whose distances from two fixed points, the foci, add up to a constant $2a$ . In the focus and directrix form it has $e < 1$ . The standard equation is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with $b^2 = a^2(1 - e^2)$ , the foci are $(\pm ae, 0)$ and the latus rectum is $\dfrac{2b^2}{a}$ . Every point is $(a\cos\theta, b\sin\theta)$ .
1 simulation1 practice question1 Python program4 problems1 Python lab
Read the lesson
Lesson 2about 30 min
The hyperbola
A hyperbola is the set of points whose distances from two foci differ by a constant $2a$ , or the set with $PF = e \cdot PM$ and $e > 1$ . The standard equation is $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ with $b^2 = a^2(e^2 - 1)$ . It has two branches, and its asymptotes are $y = \pm\dfrac{b}{a}x$ . The rectangular hyperbola has $e = \sqrt{2}$ and can be written $xy = c^2$ .
1 simulation1 practice question1 Python program2 problems
Read the lesson
Lesson 3about 30 min
Tangents, normals and the general equation of a conic
The tangent to the ellipse at $(x_1, y_1)$ is $\dfrac{xx_1}{a^2} + \dfrac{yy_1}{b^2} = 1$ and for the hyperbola it is $\dfrac{xx_1}{a^2} - \dfrac{yy_1}{b^2} = 1$ . With a given slope $m$ the tangents are $y = mx \pm \sqrt{a^2m^2 \pm b^2}$ . The general second degree equation is a conic, and the sign of $h^2 - ab$ says which one.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
Read the lesson

JEE Main format20 questions · 50 min

Not started

Loading…