Let the eccentricity of the hyperbola [H:x^2 a^2 - y^2 b^2 = 1] be [ 5 2 ] and length of i…

Question

Let the eccentricity of the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $$\sqrt {{5 \over 2}} $$ and length of its latus rectum be $$6\sqrt 2 $$. If $y = 2x + c$ is a tangent to the hyperbola H, then the value of c2 is equal to :

Accepted Answer

$$1 + {{{b^2}} \over {{a^2}}} = {5 \over 2} \Rightarrow {{{b^2}} \over {{a^2}}} = {3 \over 2}$$

$${{2{b^2}} \over a} = 6\sqrt 2  \Rightarrow 2.\,{3 \over 2}.\,a = 6\sqrt 2 $$

$$ \Rightarrow a = 2\sqrt 2 ,\,{b^2} = 12$$

$${c^2} = {a^2}{m^2} - {b^2} = 8.4 - 12 = 20$$

Solutions · 1