Question

If $\frac{x^2}{3-\alpha }+\frac{y^2}{2}=1$ represents an ellipse whose major axis is along the $x-$axis, then the range of $\alpha$ is

• A. $(1,\infty )$
• B. $(-\infty ,1)$
• C. $(-\infty ,-1)$
• D. $(3,\infty )$

Answer 1

The correct option is B $(-\infty ,1)$

Given $\frac{x^2}{3-\alpha }+\frac{y^2}{2}=1$
Comparing this with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Then $a^2=3-\alpha ,b^2=2$
As $a^2>0$, so
$3-\alpha >0\Rightarrow \alpha <3\cdots \left(1\right)$
Also, the major axis is along the $x-$axis, so
$a^2>b^2\Rightarrow 3-\alpha >2\Rightarrow \alpha <1\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$
$\alpha \in (-\infty ,1)$