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Question #40823

Difficulty: 8/10 | 🟢 Free

The Question

Let a family of lines be given by the equation $\Delta(x, y, t) = 0$ , where $t$ is a parameter, $a > 0$ , and $\Delta(x, y, t) = \begin{vmatrix} x & at^2 & 1 \\\\ y & 2at & t \\\\ 1 & -1 & 0 \end{vmatrix}$ Let $(x_c, y_c)$ be the coordinates of the envelope of this family of lines, which can be obtained by solving the system of equations $\Delta = 0$ and $\frac{\partial \Delta}{\partial t} = 0$ . Evaluate the limit: $L = \lim_{t \to 1} \frac{x_c - y_c - a}{t - 1}$

Options

A.
$0$
B.
$-4a$
C.
$-2a$
D.
$4a$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Normal to Conic|Differentiation of Determinants|Minors and Cofactors Sign Convention|Algebraic Simplification

Why This Question Is Tricky

Common Mistakes

This question tests the differentiation of parametric determinants and algebraic limit evaluation.

The primary trap lies in the 'Minors and Cofactors Sign Convention' during row-wise differentiation.

When expanding the first derivative determinant along the first row, students frequently forget the negative sign associated with the cofactor $C_{12}$ , incorrectly evaluating it as $-2at^2$ instead of $2at^2$ .

This leads to an incorrect envelope $x_c = 2a + at^2, y_c = 2at^3$ .

The final limit is specifically designed as a 0/0 indeterminate form that yields a finite, incorrect value ( $-4a$ ) if this sign error is made, acting as a lethal algebraic filter.

Quick Facts About This Question

Question Information

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