Which one of the following is correct ?
The eccentricity of the conic $$\frac{{{x^2}}}{{{a^2} + \lambda }} + \frac{{{y^2}}}{{{b^2} + \lambda }} = 1,\,\left( {\lambda \geqslant 0} \right)$$

Solution :
Equation of the given conic is an equation of ellipse $$\frac{{{x^2}}}{{{a^2} + \lambda }} + \frac{{{y^2}}}{{{b^2} + \lambda }}\,\left( {x \geqslant 0} \right)$$
$$ \Rightarrow {A^2} = {a^2} + \lambda {\text{ and }}{B^2} = {b^2} + \lambda $$
$$\eqalign{ & {\text{Eccentricity, }}e = \sqrt {1 - \frac{{{B^2}}}{{{A^2}}}} \cr & = \sqrt {1 - \frac{{{b^2} + \lambda }}{{{a^2} + \lambda }}} \cr & = \sqrt {\frac{{{a^2} + \lambda - {b^2} - \lambda }}{{{a^2} + \lambda }}} \cr & = \sqrt {\frac{{{a^2} - {b^2}}}{{{a^2} + \lambda }}} \cr} $$
$$\lambda $$ is in the denominator so, when $$\lambda $$ increases, the eccentricity decreases.