# Hyperbola Examples

Here you will learn some hyperbola examples for better understanding of hyperbola concepts.

Example 1 : If the foci of a hyperbola are foci of the ellipse $$x^2\over 25$$ + $$y^2\over 9$$ = 1. If the eccentricity of the hyperbola be 2, then its equation is :

Solution : For ellipse e = $$4\over 5$$, so foci = ($$\pm$$4, 0)

for hyperbola e = 2, so a = $$ae\over e$$ = $$4\over 2$$ = 2, b = $$2\sqrt{4-1}$$ = 2$$\sqrt{3}$$

Hence the equation of the hyperbola is $$x^2\over 4$$ – $$y^2\over 12$$ = 1

Example 2 : The eccentricity of the conjugate hyperbola to the hyperbola $$x^2-3y^2$$ = 1 is-

Solution : Equation of the conjugate hyperbola to the hyperbola $$x^2-3y^2$$ = 1 is

$$-x^2-3y^2$$ = 1 $$\implies$$ $$-x^2\over 1$$ + $$y^2\over {1/3}$$ = 1

Here $$a^2$$ = 1, $$b^2$$ = $$1\over 3$$

$$\therefore$$   eccentricity e = $$\sqrt{1 + a^2/b^2}$$ = $$\sqrt{1+3}$$ = 2

Example 3 : Find the equation of the tangent to the hyperbola $$x^2 – 4y^2$$ = 36 which is perpendicular to the line x – y + 4 = 0

Solution : Let m be the slope of the tangent, since the tangent is perpendicular to the line x – y = 0

$$\therefore$$   m$$\times$$1 = -1 $$\implies$$ m = -1

Since $$x^2-4y^2$$ = 36 or $$x^2\over 36$$ – $$y^2\over 9$$ = 1

Comparing this with $$x^2\over a^2$$ – $$y^2\over b^2$$ = 1

$$\therefore$$   $$a^2$$ = 36 and $$b^2$$ = 9

So the equation of the tangent are y = -1x $$\pm$$ $$\sqrt{36\times {-1}^2 – 9}$$

$$\implies$$ y = x $$\pm$$ $$\sqrt{27}$$ $$\implies$$ x + y $$\pm$$ 3$$\sqrt{3}$$ = 0

Example 4 : Find the asymptotes of the hyperbola $$2x^2 + 5xy + 2y^2 + 4x + 5y$$ = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.

Solution : Let $$2x^2 + 5xy + 2y^2 + 4x + 5y + k$$ = 0 be asymptotes. This will represent two straight line

so $$abc + 2fgh – af^2 – bg^2 – ch^2$$ = 0 $$\implies$$ 4k + 25 – $$25\over 2$$ – 8 – $$25\over 4$$k = 0

$$\implies$$ k = 2

$$\implies$$ $$2x^2 + 5xy + 2y^2 + 4x + 5y + 2$$ = 0 are asymptotes

$$\implies$$ (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes

and   $$2x^2 + 5xy + 2y^2 + 4x + 5y + c$$ = 0 is general equation of hyperbola.

Practice these given hyperbola examples to test your knowledge on concepts of hyperbola.