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


Two Foci and Two Directrices of the Ellipse

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

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

\(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 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 ellipse and the point S and the line ZK are the focus and directrix respectively.

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

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

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

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

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

⇒ (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 ellipse 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 ellipse 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, an ellipse has two foci and two directrices.

● The Ellipse




11 and 12 Grade Math 

From Two Foci and Two Directrices of the Ellipse 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. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 09, 25 02:37 AM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More

  2. Construction of a Circle | Working Rules | Step-by-step Explanation |

    Jul 09, 25 01:29 AM

    Parts of a Circle
    Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

    Read More

  3. Combination of Addition and Subtraction | Mixed Addition & Subtraction

    Jul 08, 25 02:32 PM

    Add and Sub
    We will discuss here about the combination of addition and subtraction. The rules which can be used to solve the sums involving addition (+) and subtraction (-) together are: I: First add

    Read More

  4. Addition & Subtraction Together |Combination of addition & subtraction

    Jul 08, 25 02:23 PM

    Addition and Subtraction Together Problem
    We will solve the different types of problems involving addition and subtraction together. To show the problem involving both addition and subtraction, we first group all the numbers with ‘+’ and…

    Read More

  5. 5th Grade Circle | Radius, Interior and Exterior of a Circle|Worksheet

    Jul 08, 25 09:55 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More