The point $(4, 1)$ undergoes the following transformations successively: (i) Reflection about the line $y = x$ (ii) Translation through a distance of $2$ units along the positive direction of the $x$ -axis (iii) Rotation through an angle $\frac{π}{4}$ about the origin in the anticlockwise direction The final position of the point is

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $(\frac{- 1}{2}, \frac{7}{2})$

Detailed Solution

Reflection in $y = x$ swaps the coordinates: $(4, 1) \to (1, 4)$ .

Translation by $2$ units along $+ x$ : $(1, 4) \to (3, 4)$ .

Rotation by $\frac{π}{4}$ : $x' = \frac{x - y}{2} = \frac{3 - 4}{2} = \frac{- 1}{2}$ and $y' = \frac{x + y}{2} = \frac{3 + 4}{2} = \frac{7}{2}$ .

✓ Check

Rotation keeps the distance from the origin:

3^{2} + 4^{2} = 25

and

\frac{1}{2} + \frac{49}{2} = 25

⚠ Common trap

(B) uses

x' = \frac{y - x}{2}

instead of

\frac{x - y}{2}

, a sign slip in the rotation formula.

Quick tip

$(x, y) \to (\frac{x - y}{2}, \frac{x + y}{2})$

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Rectangular Cartesian System — Preparation

This question tests your understanding of Rectangular Cartesian System, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

The rectangular Cartesian coordinate system locates any point in a plane by its distances from two perpendicular axes, written as an ordered pair (x, y). It is the starting point for all of coordinate geometry. This topic covers quadrants, the distance formula, the section formula (internal and external division), the midpoint formula, the area of a triangle from its vertices, and collinearity of points. It also covers special points of a triangle such as the centroid, incentre and circumcentre, and the locus of a point. A useful check is that three points are collinear exactly when the area of the triangle they form is zero. Questions here are usually formula-based and can be solved quickly with careful sign handling. Practise the Cartesian coordinate system questions below, each with a step-by-step solution.

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