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

• Definition of Hyperbola
• Standard Equation of an Hyperbola
  
• Vertex of the Hyperbola
• Centre of the Hyperbola
• Transverse and Conjugate Axis of the Hyperbola
• Two Foci and Two Directrices of the Hyperbola
• Latus Rectum of the Hyperbola
• Position of a Point with Respect to the Hyperbola
• Conjugate Hyperbola
• Rectangular Hyperbola
• Parametric Equation of the Hyperbola
• Hyperbola Formulae
• Problems on Hyperbola

11 and 12 Grade Math 

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