Consider an ellipse $\frac{x^{2}}{f (k^{2} + 2 k + 5)} + \frac{y^{2}}{f (k + 1)} = 1, (k \in R) .$ If $f (x)$ is a positive decreasing function, then

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The set of all possible values of k, for which the major axis is y-axis, is $(- \infty, 2)$

The set of all possible values of k, for which the minor axis is x-axis, is $(- \infty, \infty)$

The set of all possible values of k, for which the major axis is y-axis, is $(- \infty, \infty)$

The set of all possible values of k, for which the major axis is y-axis, is $(- 3, \infty)$

answer is A, C.

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Detailed Solution

$k + 1 < {(k + 1)}^{2} + 4$
$' f' i s (↓)$ function;
$f (k + 1) > f (k^{2} + 2 k + 5)$
So, major axis is y-axis; minor axis is x-axis

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