# 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.

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