Loading [MathJax]/jax/output/HTML-CSS/jax.js

Subscribe to our YouTube channel for the latest videos, updates, and tips.


Parametric Equation of the Hyperbola

We will learn in the simplest way how to find the parametric equations of the hyperbola.

The circle described on the transverse axis of a hyperbola as diameter is called its Auxiliary Circle.

If x2a2 - y2b2 = 1 is a hyperbola, then its auxiliary circle is x2 + y2 = a2.

Let the equation of the hyperbola be, x2a2 - y2b2 = 1  

Parametric Equation of the Hyperbola

The transverse axis of the hyperbola x2a2 - y2b2 = 1 is AA’ and its length = 2a. Clearly, the equation of the circle described on AA’ as diameter is x2 + y2 = a2 (since the centre of the circle is the centre C (0, 0) of the hyperbola).

Therefore, the equation of the auxiliary circle of the hyperbola x2a2 - y2b2 = 1 is, x2 + y2 = a2

Let P (x, y) be any point on the equation of the hyperbola be x2a2 -y2b2 = 1

Now from P draw PM perpendicular to the transverse axis of the hyperbola. Again take a point Q on the auxiliary circle x2 + y2 = a2 such that ∠CQM = 90°.

Join the point C and Q. The length of QC = a. Again, let ∠MCQ = θ. The angle ∠MCQ = θ is called the eccentric angle of the point P on the hyperbola.

Now from the right-angled  ∆CQM we get,

CQMC = cos θ          

or, a/MC  =   a/sec θ       

or, MC  = a sec θ

Therefore, the abscissa of P = MC = x = a sec θ

Since the point P (x, y) lies on the hyperbola x2a2 -y2b2 = 1 hence,

a2sec2θa2 - y2b2 = 1, (Since, x = a sec θ)

y2b2 = sec2 θ – 1

y2b2 = tan2 θ

y2 = b2 tan2 θ

y = b tan θ

Hence, the co-ordinates of P are (a sec θ, b tan θ).

Therefore, for all values of θ the point P (a sec θ, b tan θ) always lies on the hyperbola x2a2 - y2b2  = 1  

Thus, the co-ordinates of the point having eccentric angle θ can be written as (a sec θ, b tan θ). Here (a sec θ, b tan θ) are known as the parametric co-ordinates of the point P.

The equations x = a sec θ, y = b tan θ taken together are called the parametric equations of the hyperbola x2a2 - y2b2 = 1; where θ is parameter (θ is called the eccentric angle of the point P).


Solved example to find the parametric equations of a hyperbola:

1. Find the parametric co-ordinates of the point (8, 3√3) on the hyperbola 9x2 - 16y2 = 144.

Solution:     

The given equation of the hyperbola is 9x2 - 16y2 = 144

x216 - y29 = 1

x242 - y232 = 1, which is the form of x2a2 - y2b2 = 1.  

Therefore,

a2 = 42 

⇒ a = 4 and   

b2 = 32     

⇒ b = 3.

Therefore, we can take the parametric co-ordinates of the point (8, 3√3) as (4 sec θ, 3 tan θ).

Thus we have, 4 sec θ = 8      

⇒ sec θ = 2        

⇒ θ = 60°

We know that for all values of θ the point (a sec θ, b tan θ) always lies on the hyperbola x2a2 - y2b2  = 1  

Therefore, (a sec θ, b tan θ) are known as the parametric co-ordinates of the point.

Therefore, the parametric co-ordinates of the point (8, 3√3)   are (4 sec 60°, 3 tan 60°).

 

2. P (a sec θ, a tan θ) is a variable point on the hyperbola x2 - y2 = a2, and M (2a, 0) is a fixed point. Prove that the locus of the middle point of AP is a rectangular hyperbola.

Solution:        

Let (h, k) be the middle point of the line segment AM.

Therefore, h = asecθ+2a2   

⇒ a sec θ = 2(h - a)

(a sec θ)2 = [2(h - a)]2 …………………. (i)

and k = atanθ2

⇒ a tan θ = 2k

(a tan θ)2 = (2k)2 …………………. (ii)

Now form (i) - (ii), we get,

(a sec θ)2 - (a tan θ)2 = [2(h - a)]2 - (2k)2

⇒ a2(sec2 θ - tan2 θ) = 4(h - a)2 - 4k2

⇒ (h - a)2 - k2 = a24.

Therefore, the equation to the locus of (h, k) is (x - a)2 - y2 = a24, which is the equation of a rectangular hyperbola.

The Hyperbola





11 and 12 Grade Math 

From Parametric Equation of the Hyperbola to HOME PAGE




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.



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.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Average | Word Problem on Average | Questions on Average

    May 19, 25 02:53 PM

    Worksheet on Average
    In worksheet on average we will solve different types of questions on the concept of average, calculating the average of the given quantities and application of average in different problems.

    Read More

  2. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More