A hyperbola whose transverse axis is along the major axis of the conic x^23+y^24=4 and has…

Question

A hyperbola whose transverse axis is along the major axis of the conic $\dfrac{x^2}3+\dfrac{y^2}4=4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $\dfrac32$, then which of the following points does NOT lie on it?

Accepted Answer

Conic: $\dfrac{x^2}{12}+\dfrac{y^2}{16}=1$, major axis along $y$, $e=\sqrt{1-\dfrac{12}{16}}=\dfrac12$, foci $(0,\pm2)$.

Hyperbola: vertices $(0,\pm2)\Rightarrow A^2=4$ (transverse, along $y$); $e_h=\dfrac32\Rightarrow B^2=A^2(e_h^2-1)=4\left(\dfrac54\right)=5$.

$$\frac{y^2}4-\frac{x^2}5=1.$$

Checking $(5,2\sqrt3)$: $\dfrac{12}4-\dfrac{25}5=3-5=-2\ne1$ --- does not lie on it.

Solutions · 1