Q112 / 155 · WBJEE 2023 · Mathematics

Let \(f\) be a strictly decreasing function defined on \(\mathbb{R}\) such that \(f(x) > 0, \forall x \in \mathbb{R}\). Let \(\frac{x^2}{f\left(a^2+5 a+3\right)}+\frac{y^2}{f(a+15)}=1\) be an ellipse with major axis along the \(y\)-axis. The value of ' \(a\) ' can lie in the interval (s)

  • Correct answer
    A
    \((-\infty,-6)\)
  • B
    \((-6,2)\)
  • Correct answer
    C
    \((2, \infty)\)
  • D
    \((-\infty, \infty)\)

Solution

Hint : \(f \Rightarrow\) strictly decreasing function \(\forall x \in R\) \(\begin{aligned} & f(x) > 0 \Rightarrow f^{\prime}(x) < 0 \\ & \frac{x^2}{f\left(a^2+5 a+3\right)}+\frac{y^2}{f(a+15)}=1 \end{aligned}\) As major axis is \(y\)-axis \(\begin{aligned} & f(a+15) > f\left(a^2+5 a+3\right) \\ & \Rightarrow a+15 < a^2+5 a+3 \quad (\because f \text{ is decreasing) }\\ & \Rightarrow a^2+4 a-12 > 0 \\ & \Rightarrow a < -6 \text { or } a > 2 \end{aligned}\)

Hence, the correct answer is options: 'A', 'C'.

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