A hyperbola whose transverse axis is along the major axis of the conic, [x^2 3 + y^2 4 = 4…

Question

A hyperbola whose transverse axis is along the major axis of the conic, $${{{x^2}} \over 3} + {{{y^2}} \over 4} = 4$$ and has vertices at the foci of this conic. If  the eccentricity of the hyperbola is $${3 \over 2},$$ then which of the following points does NOT lie on it?

Accepted Answer

$${{{x^2}} \over {12}} + {{{y^2}} \over {16}}$$ = 1

e = $$\sqrt {1 - {{12} \over {16}}} $$ = $${1 \over 2}$$

Foci (0, 2)   \&   (0, - 2)

So, transverse axis of hyperbola

= 2b = 4

$$ \Rightarrow $$ b = 2 \& a2 = 12 (e2 - 1)

$$ \Rightarrow $$   a2 = 4$$\left( {{9 \over 4} - 1} \right)$$

$$ \Rightarrow $$   a2 = 5

$$ \therefore $$    It's equation is $${{{x^2}} \over 5} - {{{y^2}} \over 4}$$ = - 1

The point (5, 2$$\sqrt 3 $$) does not satisfy the above equation.

Solutions · 1