# 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