Question

A hyperbola whose transverse axis is along the major axis of the conic,`x^2/3+y^2/4= 4` and has vertices atthe foci of this conic. If the eccentricity of the hyperbola is `3/2`, then which of the following points does NOT lie on it?
A. `(sqrt(5),2sqrt(2))`
B. `(sqrt(5),2sqrt(3))`
C. `(0,2)`
D. `(sqrt(10),2sqrt(3))`

Answer 1

The ellipse `(x^(2))/(3)+(y^(2))/(4)=4`, or `(x^(2))/(12)+(y^(2))/(16)=1` has its major axis along `y`-axis and eccentricity `e=sqrt(1-(12)/(16))=(1)/(2)`. So, coordinates of its foci are `(0,2)` and `(0,-2)` . The vertices of the hyperbola are at `(0,2)` and `(0,-2)` .So, the length of its transverse axis is `2b=4`. Let the length of its conjugate axis be `2a`. Then,
`a^(2)=b^(2)(e^(2)-1)impliesa^(2)=4((9)/(4)-1)=5`
So, the equation of the hyperbola is `(x^(2))/(5)-(y^(2))/(4)=-1`
Clearly, `(5,2sqrt(3))` does not lie on this hyperbola.