Question

Consider the ellipse $\frac{x^2}{f(k^2+3k+5)}+\frac{y^2}{f(k+13)}=1$. If $f\left(x\right)$ is positive decreasing function, then

• A. Number of positive integral values of $k$ for which $x$ axis is major axis equals $1$
• B. the set of values of $k$ for which t$y-$axis is the major axis is $(-\infty ,2)$
• C. the set of values of $k$ for which $y-$axis is the major axis is $(-\infty ,-4)\cup (2,\infty )$
• D. the set of values of $k$ for which $y-$axis is the major axis is $(-4,\infty )$

Answer 1

Answer: (A), (C)

The correct options are
A Number of positive integral values of $k$ for which $x$ axis is major axis equals $1$
C the set of values of $k$ for which $y-$axis is the major axis is $(-\infty ,-4)\cup (2,\infty )$

For $x-$ axis is be major axis
$f(k^2+3k+5)>f(k+13)k^2+3k+5<k+13k^2+2k-8<0k\in (-4,2)$
and for rest of the values of $k$ major axis will be $y-$ axis