Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
Consider the ellipse $\frac{x^2}{t(k^2 + 2k + 5)} + \frac{y^2}{t(k+1)} = 1$ and $f(x)$ is a positive decreasing function, then:
The set of values of $k$, for which the major axis is $x$-axis is $(-3, 2)$
The set of values of $k$, for which the major axis is $y$-axis is $(-\infty, 2)$
The set of values of $k$, for which the major axis is $y$-axis is $(-\infty, -3) \cup (2, \infty)$
The set of values of $k$, for which the major axis is $y$-axis is $(-3, \infty)$
Step-by-Step Solution
Key Concept: For a decreasing function, a larger function value corresponds to a smaller input, so reverse the inequality when comparing arguments.
For $f(x)$ to be decreasing with major axis along the x-axis, the condition $f(k^2+2k+5) > f(k+11)$ implies $k^2 + 2k + 5 < k + 11$, giving $k \in (-3, 2)$. For all other values of $k$, namely $k \in (-\infty, -3) \cup (2, \infty)$, the major axis must be along the y-axis.
Correct Answer: 1,3