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Question #40788

Difficulty: 8/10 | 🟢 Free

The Question

Two fixed points $A$ and $B$ have coordinates $(3, 0)$ and $(-3, 0)$ respectively in a fixed Cartesian coordinate system $xy$ . A variable point $P(x,y)$ moves such that $PA + PB = 10$ . A new coordinate system $XY$ is defined by shifting the origin to $O'(t, 2t)$ , where $t$ is a parameter. If the parameter $t$ is continuously adjusted such that it always equals the $Y$ -coordinate of $P$ in the new system (i.e., $t = Y$ ), which of the following represents the locus of $P$ in the new $XY$ coordinate system?

Options

A.
$16X^2 + 32XY + 241Y^2 = 400$
B.
$16X^2 - 32XY + 41Y^2 = 400$
C.
$16X^2 + 32XY + 91Y^2 = 400$
D.
$16X^2 + 25Y^2 = 400$

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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

Shifting Origin|Locus via Distance Constraints|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-inertial coordinate transformations where the shift parameter is coupled to the moving point's own coordinates.

The common trap is assuming the locus shape is invariant under such a shift (leading to $16X^2 + 25Y^2 = 400$ ) or incorrectly applying the shift formula as $x = X - t$ instead of $x = X + t$ (leading to $16X^2 - 32XY + 41Y^2 = 400$ ).

By linking the parameter $t$ to $Y$ , the transformation becomes a shear mapping, fundamentally altering the standard ellipse equation and forcing a rigorous coordinate substitution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Shifting Origin
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free