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Question #40930

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the function representing the rectangular hyperbola $xy = 8$ in the first quadrant. Let $P(x_0, y_0)$ be a point on this hyperbola such that $\lim_{x \to x_0} \frac{x^{10} (f(x))^8 - x_0^{10} (f(x_0))^8}{x^5 (f(x))^9 - x_0^5 (f(x_0))^9} = -256$ Let $N$ be the normal to the hyperbola at the point $P$ . If $N$ intersects the coordinate axes at points $A$ and $B$ , find the area of the triangle $OAB$ , where $O$ is the origin.

Options

A.
9
B.
12
C.
16
D.
18

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

Algebraic Simplification|Rectangular Hyperbola|Tangent/Normal Slope Interplay

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize hidden algebraic invariants before applying calculus techniques.

The primary trap is the immediate application of L'Hopital's rule to a complex rational function involving high powers of $x$ and $f(x)$ .

Doing so results in a convoluted derivative requiring the product rule and implicit differentiation, which is highly prone to algebraic errors under exam conditions.

The elegant, 'Advanced' path requires the student to use the rectangular hyperbola's property $x f(x) = c^2$ to factor out constants, drastically simplifying the limit into a manageable form.

Furthermore, it tests the interplay between the tangent slope $f'(x_0) = -y_0/x_0$ and the normal slope, culminating in a coordinate geometry application.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Algebraic Simplification
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free