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.
