A hyperbola has its centre at the origin, passes through the point (4,2) and has transverse axis of length 4 along the x-axis. Then the eccentricity of the hyperbola is:
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Official solution
Let the equation of hyperbola
\[{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}}\] = 1
Given 2a = 4
\[ \Rightarrow \] a = 2
It passes through (4, 2)
\[ \therefore \] \[{{16} \over 4} - {4 \over {{b^2}}}\] = 1
\[ \Rightarrow \] b2 = \[{4 \over 3}\]
e = \[\sqrt {1 + {{{b^2}} \over {{a^2}}}} \] = \[\sqrt {1 + {{4/3} \over 4}} \]
= \[\sqrt {1 + {1 \over 3}} \] = \[{2 \over {\sqrt 3 }}\]