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

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?