Hyperbola Examples

HYPERBOLA EXAMPLES

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.


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