Ellipse Notes

Introduction

An ellipse is a fundamental concept in conic sections and is crucial for the JEE Main Mathematics syllabus. Understanding ellipses involves grasping their geometric properties, algebraic representation, and various applications. This study note will break down the concept of ellipses into digestible parts, ensuring a comprehensive understanding.

Definition and Basic Properties

An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points (called foci) is constant. This constant is greater than the distance between the foci.

Standard Form of an Ellipse

The standard form of an ellipse centered at the origin can be written as:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

where:

• $a$ is the semi-major axis,
• $b$ is the semi-minor axis,
• $a > b$.

Important Points and Terms

1. Center: The midpoint of the line segment joining the foci, usually at the origin $(0,0)$.
2. Vertices: The endpoints of the major axis, located at $(\pm a, 0)$.
3. Co-vertices: The endpoints of the minor axis, located at $(0, \pm b)$.
4. Foci: The points $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$.

Eccentricity

The eccentricity $e$ of an ellipse is given by:

$$ e = \frac{c}{a} $$

where $0

< e < 1$.

Derivation of the Equation of an Ellipse

To derive the standard form of an ellipse, consider the definition: the sum of the distances from any point $(x, y)$ on the ellipse to the foci $(\pm c, 0)$ is constant and equals $2a$.

Using the distance formula, we get:

$$ \sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a $$

Solving this equation involves algebraic manipulation and squaring both sides, which eventually leads to the standard form.

Parametric Equations

The parametric equations of an ellipse are useful for plotting and understanding its geometry. For an ellipse centered at the origin, the parametric equations are:

$$ x = a \cos \theta \ y = b \sin \theta $$

where $\theta$ ranges from $0$ to $2\pi$.

Tangents and Normals

Equation of Tangent

The equation of the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at a point $(x_1, y_1)$ is:

$$ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 $$

Equation of Normal

The equation of the normal to the ellipse at $(x_1, y_1)$ is more complex but can be derived using calculus. It is given by:

$$ a^2 x x_1 - b^2 y y_1 = a^2 x_1^2 - b^2 y_1^2 $$

Ellipse in Different Orientations

Horizontal and Vertical Ellipses

The standard form discussed so far is for a horizontally oriented ellipse. For a vertically oriented ellipse, the equation is:

$$ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 $$

General Form

The general form of an ellipse can be written as:

$$ Ax^2 + By^2 + Cx + Dy + E = 0 $$

where $A$ and $B$ are positive constants. This form can be converted to the standard form by completing the square.

Applications

Ellipses have various applications in physics, engineering, and astronomy. For instance, the orbits of planets and satellites are elliptical.

Summary

• An ellipse is defined by the sum of distances from two foci being constant.
• The standard form is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
• Key properties include the center, vertices, co-vertices, foci, and eccentricity.
• Parametric equations and the equations of tangents and normals are crucial for solving problems.
• Different orientations and general forms broaden the understanding of ellipses.

By mastering these concepts, one can solve a wide range of problems involving ellipses in the JEE Main Mathematics syllabus.