Before You Begin

Question #40793

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ( $a > b > 0$ ) with eccentricity $e$ . The value of $e^2$ is given by the limit: $e^2 = \lim_{u \to 0} \frac{u^3 \sin\left(\frac{1}{u}\right) + \ln\left(1 + \frac{3}{4}u^2\right)}{u^2}$ Let $f(t) = (t^2 - 5t + 5)^{\ln(t^2 - 5t + 5)}$ and $S$ be the set of all critical points of $f(t)$ in its domain. Let $P(x_0, y_0)$ be a point on $C$ in the first quadrant. The normal to $C$ at $P$ intersects the x-axis at $G(X_N, 0)$ , where $X_N$ is the sum of all elements in $S$ . If the tangent to $C$ at $P$ intersects the x-axis at $T(X_T, 0)$ with $X_T = 12$ , find the value of $9y_0^2$ .

Options

A.
80
B.
45
C.
60
D.
75

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

General Conic Definition|Normal to Conic|Logarithmic Differentiation Domain|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question deploys a 'Lethal Collision' between calculus and coordinate geometry.

First, it tests the 'L'Hopital Pitfall' (Concept 161) in the eccentricity limit; blindly applying L'Hopital's rule to $u^3 \sin(1/u) / u^2$ yields an oscillating, non-existent limit, whereas the standard squeeze theorem correctly evaluates it to 0.

Second, it uses the 'Logarithmic Differentiation Domain' (Concept 250) as a hard filter for the normal's intercept.

When finding the critical points of $f(t)$ , students algebraically obtain $t=2.5, 1, 4$ .

However, $t=2.5$ yields a negative base for the logarithmic exponent, violating the domain.

Failing to reject $t=2.5$ leads to an extraneous intercept $X_N = 7.5$ , cascading into the trap answer 45.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	General Conic Definition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free