A hyperbola having the transverse axis of length √2 has the same foci as that of the ellip…

Question

A hyperbola having the transverse axis of length $\sqrt2$ has the same foci as that of the ellipse $3x^2+4y^2=12$. Then this hyperbola does not pass through which of the following points?

Accepted Answer

Ellipse: $\dfrac{x^2}4+\dfrac{y^2}3=1$, $c=\sqrt{4-3}=1$, foci $(\pm1,0)$.

Hyperbola: $2a=\sqrt2\Rightarrow a=\dfrac1{\sqrt2}$; $c^2=a^2+b^2\Rightarrow1=\dfrac12+b^2\Rightarrow b^2=\dfrac12$.

$2x^2-2y^2=1.$

Checking $\left(\frac32,\frac1{\sqrt2}ight)$: $2\left(\frac94ight)-2\left(\frac12ight)=\frac92-1=\frac72
e1$ --- does not lie on it.

Solutions · 1