Definition of Hyperbola

We will discuss the definition of hyperbola and how to find the equation of the hyperbola whose focus, directrix and eccentricity are given.

If a point (P) moves in the plane in such a way that the ratio of its distance from a fixed point (S is known as focus) in the same plane to its perpendicular distance from the fixed line (L is known as dirctrix) is always constant which is always greater than unity, then the locus traced out by P is called a Hyperbola.

Let S and L be a fixed point and a fixed straight line on a plane respectively. If the point P moves on this plane in such a way that its distance from the fixed point S always constant ratio to its perpendicular distance from the fixed line L and its ratio is greater than unity then the locus of the point P is called a Hyperbola.

Definition of Hyperbola

The fixed point S is called a focus and the fixed straight line L, the corresponding directrix and the constant ratio is called the eccentricity of the hyperbola. The eccentricity is generally denoted by e (> 1).

If S is the focus, Z is the directrix and P is any point on the hyperbola, then by definition

\(\frac{SP}{PM}\) = e

⇒ SP = e PM


Solved example to find the equation of the hyperbola whose focus, directrix and eccentricity are given:

The equation of the directrix of a hyperbola is x + y = -1. Its focus is at (1, 2) and the eccentricity is \(\frac{3}{2}\). Find the equation of the hyperbola.

Solution:

Let P(x, y) be any point on the required hyperbola. If PM is the length of the perpendicular from P upon the directrix  x + y = -1 or, x + y + 1 = 0 then

PM = \(\frac{x + y + 1}{\sqrt{1^{2} + (-1)^{2}}}\) =  \(\frac{x + y + 1}{√2}\)        

Again, the distance of P from the focus S (- 1, 1) is

SP = \(\sqrt{(x - 1)^2 + (y - 1)^2}\)

Since the point Plies on the required hyperbola, hence by definition we have,

\(\frac{SP}{PM}\) = e

SP = e PM

⇒ SP\(^{2}\) = e\(^{2}\)(PM)\(^{2}\)

⇒ (x - 1)\(^{2}\) + (y - 2)\(^{2}\) = \(\frac{9}{4}\) \(\frac{(x + y + 1)^{2}}{2}\), [Since, e = 3]

⇒ 8x\(^{2}\) + 8y\(^{2}\) - 16x - 32y + 40 = 9x\(^{2}\) + 9y\(^{2}\) + 9 + 18xy + 18x + 18y

⇒ x\(^{2}\) + y\(^{2}\) + 18xy + 34x + 50y - 31 = 0, which is the required equation of the hyperbola.

The Hyperbola





11 and 12 Grade Math 

From Definition of Hyperbola to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Multiplication of a Number by a 3-Digit Number |3-Digit Multiplication

    Mar 28, 24 11:44 AM

    Multiplying by 3-Digit Number
    In multiplication of a number by a 3-digit number are explained here step by step. Consider the following examples on multiplication of a number by a 3-digit number: 1. Find the product of 36 × 137

    Read More

  2. Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

    Mar 27, 24 05:21 PM

    Multiply 2-Digit Numbers by a 2-Digit Numbers
    How to multiply a number by a 2-digit number? We shall revise here to multiply 2-digit and 3-digit numbers by a 2-digit number (multiplier) as well as learn another procedure for the multiplication of…

    Read More

  3. Multiplication by 1-digit Number | Multiplying 1-Digit by 4-Digit

    Mar 26, 24 04:14 PM

    Multiplication by 1-digit Number
    How to Multiply by a 1-Digit Number We will learn how to multiply any number by a one-digit number. Multiply 2154 and 4. Solution: Step I: Arrange the numbers vertically. Step II: First multiply the d…

    Read More

  4. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Mar 25, 24 05:36 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  5. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Mar 25, 24 04:18 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More