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  4. The equation -(x2/2-λ)-(y2/λ-5)-1=0 represents an ellipse if

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

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Solution:

$-\frac{x^{2}}{2-\lambda}-\frac{y^{2}}{\lambda-5}-1=0$
$\frac{-x^{2}}{(2-\gamma)}-\frac{y^{2}}{(\lambda-5)}=1$
$\frac{x^{2}}{\lambda-2}+\frac{y^{2}}{5-\lambda}=1$
if represents an ellipse if $\lambda-2 > 0\, \&\, 5-\lambda > 0$
$\lambda > 2\, \&\, 5 > \lambda$
$\lambda < 2\, \&\, \lambda < 5$
$\therefore 2 < \lambda < 5$

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