Question

The equation $\frac{x^2}{2-\mathrm{\lambda }}+\frac{y^2}{\mathrm{\lambda }-5}+1=0$ represents an ellipse, if
(a) λ < 5
(b) λ < 2
(c) 2 < λ < 5
(d) λ < 2 or λ > 5

Answer 1

$$\begin{gathered} \left(\mathrm{c}\right)2<\lambda <5 \\ \frac{x^2}{2-\lambda }+\frac{y^2}{\lambda -5}+1=0 \\ \Rightarrow \frac{x^2}{\lambda -2}+\frac{y^2}{5-\lambda }=1 \\ \text{To represent the equation of ellipse, we have:} \\ \lambda -2>0 \\ \Rightarrow \lambda >2 \\ \mathrm{and} \\ 5-\lambda >0 \\ \Rightarrow 5<\lambda \\ \text{∴}2<\lambda <5 \end{gathered}$$