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

Hyperbola: Standard Equation and Basic Properties

Hello students! Welcome to the world of hyperbolas, a fascinating curve with many applications, from optics to satellite trajectories. For the JEE Main, understanding the hyperbola and its properties is crucial for scoring well in coordinate geometry. Let's dive in!

1. Definition: The Difference of Distances

Imagine two fixed points, called foci (plural of focus). A hyperbola is the set of all points in a plane such that the difference of their distances from the two foci is a constant. This constant is usually denoted as $2a$ .

Geometrically, picture this: if you take any point on the hyperbola and measure its distance to each focus, the difference between those two distances will always be the same value ( $2a$ ). This is what defines the shape of the hyperbola. This geometric definition leads to the algebraic equation.

2. Standard Form of the Hyperbola

The standard form of a hyperbola centered at the origin with foci on the x-axis is given by:

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

Here, $a$ and $b$ are constants that determine the shape and size of the hyperbola. The minus sign is what distinguishes it from an ellipse (which has a plus sign). The point $(x, y)$ represents any point on the hyperbola.

3. Axes, Vertices, and Foci

Let's break down the anatomy of a hyperbola:

Transverse Axis: The line segment connecting the two vertices. Its length is $2a$ . In the standard form above, it lies along the x-axis.
Conjugate Axis: The line segment perpendicular to the transverse axis, passing through the center of the hyperbola. Its length is $2b$ . In the standard form above, it lies along the y-axis.
Vertices: The points where the hyperbola intersects the transverse axis. For the standard form, the vertices are located at $(\pm a, 0)$ . These are the "turning points" of the hyperbola.
Foci: The two fixed points used in the definition of the hyperbola. For the standard form, the foci are located at $(\pm ae, 0)$ , where $e$ is the eccentricity (more on that below).

Imagine the hyperbola "opening up" from the vertices, extending towards infinity. The foci are always located inside the curves of the hyperbola.

4. Eccentricity

Eccentricity ( $e$ ) is a measure of how "stretched" the hyperbola is. It is defined as:

e = \sqrt{1 + \frac{b^2}{a^2}}

For a hyperbola, $e > 1$ . The larger the value of $e$ , the more "open" the hyperbola is. Notice that if $b=0$ , then $e=1$ , which is the limiting case of a parabola. As $b$ gets larger compared to $a$ , $e$ increases, and the hyperbola becomes wider.

Another useful formula relating $a$ , $b$ , and $e$ is:

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

This formula is simply a rearrangement of the eccentricity formula and is useful for finding $b$ if you know $a$ and $e$ , or vice versa.

Intuition: Eccentricity tells us how much the hyperbola deviates from being a circle (which has $e=0$ ). A high eccentricity means a significant deviation, resulting in a more elongated shape.

5. Directrix and Latus Rectum

Directrix: A line such that the ratio of the distance of a point on the hyperbola from the focus to its distance from the directrix is equal to the eccentricity. For the standard hyperbola, the directrices are vertical lines given by:

x = \pm \frac{a}{e}

There are two directrices, one on each side of the center. For any point $P$ on the hyperbola, the ratio $\frac{\text{distance from } P \text{ to focus}}{\text{distance from } P \text{ to directrix}} = e$ .

Latus Rectum: A line segment passing through a focus, perpendicular to the transverse axis, with endpoints on the hyperbola. The length of the latus rectum is:

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

There are two latera recta (plural of latus rectum), one for each focus. The latus rectum helps define the "width" of the hyperbola at the foci.

Tip: Memorize the standard form equations and the formulas for foci, vertices, directrices, and latus rectum. They are frequently used in JEE problems.

Common Mistake: Confusing the formulas for the hyperbola with those for the ellipse. The sign between the $x^2$ and $y^2$ terms is crucial!

Geometric Interpretation of Directrix: The directrix is related to the focus in defining the hyperbola. The eccentricity $e$ is the key: for any point on the hyperbola, its distance to the focus is $e$ times its distance to the directrix. This relationship is fundamental to the hyperbola's shape.

JEE-Specific Trick: Many JEE problems involve finding the equation of a tangent or normal to a hyperbola. Remember the standard forms and use the properties of foci and directrices to simplify the problem. Also, remember the relationship $b^2 = a^2(e^2 - 1)$ as it can often help you quickly eliminate answer choices.

Let's consider an example to illustrate these concepts. Suppose we have a hyperbola with the equation $\frac{x^2}{9} - \frac{y^2}{16} = 1$ . Here, $a^2 = 9$ and $b^2 = 16$ , so $a = 3$ and $b = 4$ . The eccentricity is $e = \sqrt{1 + \frac{16}{9}} = \sqrt{\frac{25}{9}} = \frac{5}{3}$ . The foci are $(\pm ae, 0) = (\pm 5, 0)$ , and the vertices are $(\pm a, 0) = (\pm 3, 0)$ . The directrices are $x = \pm \frac{a}{e} = \pm \frac{9}{5}$ , and the latus rectum is $\frac{2b^2}{a} = \frac{32}{3}$ . Understanding these values provides a comprehensive understanding of the hyperbola’s geometry.

By mastering these concepts and formulas, you'll be well-prepared to tackle hyperbola problems in the JEE Main. Keep practicing, and you'll ace it! Good luck!