Standard Equation and Basic Properties | JEE Main Coordinate Geometry Crash Course | Mathematicon

Ellipse: Standard Equation and Basic Properties

Hello JEE aspirants! The ellipse is a crucial conic section, frequently tested in JEE Main. Mastering its standard equation and basic properties will not only help you solve problems directly related to ellipses but also provide a solid foundation for understanding other conic sections and their applications. Let's dive in!

Definition of an Ellipse

An ellipse is defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed points (called the foci) is constant. This constant sum is equal to $2a$ , where $a$ is a parameter related to the ellipse's size.

Imagine two pins (foci) on a board, a loop of string longer than the distance between the pins, and a pencil holding the string taut. As you move the pencil around, keeping the string tight, you'll trace out an ellipse!

Standard Form of the Ellipse Equation

The standard equation of an ellipse, where the major axis lies along the x-axis and the center is at the origin, is given by:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ (a > b)}

Here, $a$ and $b$ are positive constants, with $a > b$ . This condition ensures that the ellipse is elongated along the x-axis. If $a < b$ , the ellipse would be elongated along the y-axis. The larger value is always associated with the major axis. The values $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Major Axis, Minor Axis, Vertices, and Foci

Major Axis: The longest diameter of the ellipse, passing through the center and the foci. Its length is $2a$ . In the standard equation, it lies along the x-axis.
Minor Axis: The shortest diameter of the ellipse, passing through the center and perpendicular to the major axis. Its length is $2b$ . In the standard equation, it lies along the y-axis.
Vertices: The endpoints of the major axis. In the standard equation, the vertices are at $(\pm a, 0)$ . These are the points on the ellipse farthest from the center.
Foci: The two fixed points used in the definition of the ellipse. Their location is crucial in determining the shape of the ellipse.

Eccentricity

Eccentricity, denoted by $e$ , is a measure of how "stretched" an ellipse is. It's defined as:

e = \sqrt{1 - \frac{b^2}{a^2}}, \text{ where } 0 < e < 1

Notice that $e$ is always between 0 and 1. When $e$ is close to 0, the ellipse is nearly circular. As $e$ approaches 1, the ellipse becomes more elongated. A circle is a special case of an ellipse where $a = b$ and $e = 0$ . We can rearrange the equation to relate a, b and e as follows:

b^2 = a^2(1 - e^2)

The foci are located at a distance of $ae$ from the center along the major axis.

\text{Foci: } (\pm ae, 0), \text{ Vertices: } (\pm a, 0)

Geometric Interpretation: The eccentricity tells us how far the foci are from the center relative to the length of the semi-major axis. A higher eccentricity means the foci are farther from the center, making the ellipse more elongated.

Directrix and Latus Rectum

Now let's discuss two more key properties.

The directrix of an ellipse is a line such that the ratio of the distance from any point on the ellipse to the focus and to the directrix is equal to the eccentricity $e$ .

\text{Directrix: } x = \pm \frac{a}{e}

Geometric Interpretation: The directrices are two vertical lines, one to the left and one to the right of the ellipse. The closer the directrix is to the center, the higher the eccentricity is, making the ellipse skinnier.

The latus rectum is a line segment passing through a focus of the ellipse, perpendicular to the major axis, with endpoints on the ellipse.

\text{Latus rectum: } \frac{2b^2}{a}

Derivation: To find the length of the latus rectum, substitute $x = ae$ into the ellipse equation and solve for $y$ . This will give the y-coordinate of the endpoints of the latus rectum. The length of the latus rectum is twice this value.

Tips for Solving Problems

Tip 1: Always identify $a$ and $b$ correctly. Remember, $a$ is always greater than $b$ in the standard form we discussed ( $a > b$ ).

Tip 2: When given information about the sum of the distances from foci, remember that it equals $2a$ .

Tip 3: Use the relationship $b^2 = a^2(1 - e^2)$ to find $e$ if $a$ and $b$ are known, or to find $b$ if $a$ and $e$ are known.

Common Mistakes to Avoid

Mistake 1: Confusing $a$ and $b$ . Always ensure that $a > b$ when using the standard equation.

Mistake 2: Incorrectly applying the eccentricity formula. Double-check that you are subtracting $\frac{b^2}{a^2}$ from 1, not the other way around.

Mistake 3: Forgetting the $\pm$ sign when calculating the foci $(\pm ae, 0)$ and the directrix $x = \pm \frac{a}{e}$ . There are two foci and two directrices!

JEE-Specific Tricks

Trick 1: In JEE problems, ellipses are often combined with other concepts like tangents, normals, and parametric equations. Practice problems involving these combinations.

Trick 2: Look out for problems where you are given the foci and a point on the ellipse. Use the definition of the ellipse (sum of distances from foci is constant) to find the equation.

Keep practicing, and you'll master the ellipse in no time! Good luck with your JEE preparation!