Question

The equation $\frac{x^2}{2-\lambda }+\frac{y^2}{\lambda -5}+1=0$ represents an ellipse, if

• A. $\lambda \in (-\infty ,2)\cup (5,\infty )$
• B. $\lambda \in (2,5)$
• C. $\lambda \in (2,\frac{7}{2})\cup (\frac{7}{2},5)$
• D. $\lambda \in (2,\frac{7}{2})\cup (5,\infty )$

Answer 1

The correct option is C $\lambda \in (2,\frac{7}{2})\cup (\frac{7}{2},5)$

$\frac{x^2}{2-\lambda }+\frac{y^2}{\lambda -5}+1=0$
$\Rightarrow \frac{x^2}{\lambda -2}+\frac{y^2}{5-\lambda }=1$
To represent the equation of ellipse,
$\lambda -2>0$
$\Rightarrow \lambda >2\cdots \left(i\right)$
and $5-\lambda >0$
$\Rightarrow \lambda <5\cdots \left(ii\right)$
Also, $\lambda -2\ne 5-\lambda$
$\Rightarrow \lambda \ne \frac{7}{2}\dots \left(iii\right)$

From $\left(i\right),\left(ii\right)$ and $\left(iii\right),$ we get
$\lambda \in (2,\frac{7}{2})\cup (\frac{7}{2},5)$