Two Foci and Two Directrices of the Hyperbola

We will learn how to find the two foci and two directrices of the hyperbola.

Let P (x, y) be a point on the hyperbola.

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

⇒ b\(^{2}\)x\(^{2}\) - a\(^{2}\)y\(^{2}\) = a\(^{2}\)b\(^{2}\)

Now form the above diagram we get,

CA = CA' = a and e is the eccentricity of the hyperbola and the point S and the line ZK are the focus and directrix respectively.

Two Foci and Two Directrices of the Hyperbola

Now let S' and K' be two points on the x-axis on the side of C which is opposite to the side of S such that CS' = ae and CK' = \(\frac{a}{e}\).

Further let Z'K' perpendicular CK' and PM' perpendicular Z'K' as shown in the given figure. Now join P and S'. Therefore, we clearly see that PM’ = NK'.

Now from the equation b\(^{2}\)x\(^{2}\) - a\(^{2}\)y\(^{2}\) = a\(^{2}\)b\(^{2}\), we get,

a\(^{2}\)(e\(^{2} - 1\)) x\(^{2}\) - a\(^{2}\)y\(^{2}\) = a\(^{2}\) ∙  a\(^{2}\)(e\(^{2} - 1\)), [Since, b\(^{2}\) = a\(^{2}\)(e\(^{2} - 1\))]

x\(^{2}\)(e\(^{2} - 1\)) - y\(^{2}\) = a\(^{2}\)(e\(^{2} - 1\)) = a\(^{2}\)e\(^{2}\) - a\(^{2}\)

x\(^{2}\)e\(^{2}\) - x\(^{2}\) - y\(^{2}\) = a\(^{2}\)e\(^{2}\) - a\(^{2}\)

x\(^{2}\)e\(^{2}\) + a\(^{2}\) + 2 xe a = x\(^{2}\) + a\(^{2}\)e\(^{2}\) + 2 x ae x  + y\(^{2}\)

(ex + a)\(^{2}\) = (x + ae)\(^{2}\) + y\(^{2}\)

(x + ae)\(^{2}\) + y\(^{2}\) = (ex + a)\(^{2}\)

⇒  (x + ae)\(^{2}\) - (y - 0)\(^{2}\) = e\(^{2}\)(x + \(\frac{a}{e}\))\(^{2}\)

S'P\(^{2}\) = e\(^{2}\) PM'\(^{2}\)

S'P = e PM'

Distance of P from S' = e (distance of P from Z'K')

Hence, we would have obtained the same curve had we started with S' as focus and Z'K' as directrix. This shows that the hyperbola has a second focus S' (-ae, 0) and a second directrix x = -\(\frac{a}{e}\).

In other words, from the above relation we see that the distance of the moving point P (x, y) from the point S' (- ae, 0) bears a constant ratio e (> 1) to its distance from the line x + \(\frac{a}{e}\) = 0.

Therefore, we shall have the same hyperbola if the point S' (- ae, 0) is taken as the fixed point i.e, focus and x + \(\frac{a}{e}\) = 0 is taken as the fixed line i.e., directrix.

Hence, a hyperbola has two foci and two directrices.

The Hyperbola

11 and 12 Grade Math 

From Two Foci and Two Directrices 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. Successor and Predecessor | Successor of a Whole Number | Predecessor

    May 24, 24 06:42 PM

    Successor and Predecessor of a Whole Number
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More

  2. Counting Natural Numbers | Definition of Natural Numbers | Counting

    May 24, 24 06:23 PM

    Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

    Read More

  3. Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

    May 24, 24 06:22 PM

    The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

    Read More

  4. Math Questions Answers | Solved Math Questions and Answers | Free Math

    May 24, 24 05:37 PM

    Math Questions Answers
    In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students

    Read More

  5. Estimating Sum and Difference | Reasonable Estimate | Procedure | Math

    May 24, 24 05:09 PM

    Estimating Sum or Difference
    The procedure of estimating sum and difference are in the following examples. Example 1: Estimate the sum 5290 + 17986 by estimating the numbers to their nearest (i) hundreds (ii) thousands.

    Read More