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Question #41217

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be the curve defined implicitly by $3xy - 9x - 4y^2 + 18y - 23 = 0$ . Let $O'$ be the point of intersection of the asymptotes of $C$ . A point $P(x_0, y_0)$ with $y_0 > 3$ lies on $C$ such that the normal to $C$ at $P$ passes through $O'$ . If $L_1$ is the tangent to $C$ at $P$ , and $L_2$ is a line parallel to $L_1$ passing through the origin $(0,0)$ , let $d$ be the perpendicular distance between $L_1$ and $L_2$ . Find the value of $10d^2$ .

Options

A.
324
B.
361
C.
400
D.
289

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Derivative of Implicit Inverse Forms|Condition of Collinearity|Shifting Origin|Distance Between Parallel Lines

Why This Question Is Tricky

Common Mistakes

This question tests the strategic use of coordinate translation to simplify implicit differentiation and geometric conditions.

The 'Lethal Collision' occurs if students attempt to find the normal directly from the unshifted implicit equation $3xy - 9x - 4y^2 + 18y - 23 = 0$ .

Equating the unshifted normal slope to the collinearity condition $\frac{y-3}{x-2}$ generates a complex hyperbola that must be solved simultaneously with the original curve, leading to severe algebraic exhaustion.

Shifting the origin to the intersection of the asymptotes $O'(2,3)$ decouples the variables, transforming a grueling $O(n^2)$ algebraic system into a trivial polynomial $25Y^4 = 25$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Derivative of Implicit Inverse Forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free