Question

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

Options

•  λ < 5

• λ < 2

• 2 < λ < 5

• λ < 2 or λ > 5

Answer 1

\[2 < \lambda < 5\]

\[\frac{x^2}{2 - \lambda} + \frac{y^2}{\lambda - 5} + 1 = 0\]

\[ \Rightarrow \frac{x^2}{\lambda - 2} + \frac{y^2}{5 - \lambda} = 1\]

To represent the equation of ellipse, we have:

\[\lambda - 2 > 0\]

\[ \Rightarrow \lambda > 2\]

and

\[5 - \lambda > 0\]

\[ \Rightarrow 5 < \lambda\]

\[\therefore2 < \lambda < 5\]