Straight Line - JEE Advanced Previous Year Questions with Solutions

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Q. Let $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ be the points on the plane with position vectors $-2 \hat{\mathrm{i}}-\hat{\mathrm{j}}, 4 \hat{\mathrm{i}}, 3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}$ and $-3 \hat{\mathrm{j}}$ and $-3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}$ respectively. The quadrilateral PQRS must be a (A) parallelogram, which is neither a rhombus nor a rectangle (B) square (C) rectangle, but not a square (D) rhombus, but not a square [JEE 2010, 3]

Ans. (A) $\Rightarrow$ PQRS is a parallelogram but neither a rhombus nor a rectangle.

Q. A straight line L through the point $(3,-2)$ is inclined at an angle $60^{\circ}$ to the line $\sqrt{3} x+y=1$. If $L$ also intersect the x-axis, then the equation of $L$ is (A) $y+\sqrt{3} x+2-3 \sqrt{3}=0$ (B) $\mathrm{y}-\sqrt{3} \mathrm{x}+2+3 \sqrt{3}=0$ (C) $\sqrt{3} y-x+3+2 \sqrt{3}=0$ (D) $\sqrt{3} y+x-3+2 \sqrt{3}=0$ [JEE 2011, 3 (–1)]

Ans. (B)

Q. For a point $P$ in the plane, let $d_{1}(P)$ and $d_{2}(P)$ be the distances of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2 \leq d_{1}(P)+d_{2}(P) \leq 4,$ is [JEE(Advanced)-2014, 3]

Ans. 6

Expert Tips for Straight Line in JEE Advanced

JEE Advanced rarely asks direct formula-based problems from Straight Lines. Most questions involve 2–3 concepts layered together — for example, the angle between lines combined with a constraint on intersection. Always read the question for hidden conditions like "intersects the x-axis" or "a > b > c > 0" — these constraints eliminate cases and are the actual test of your understanding.

How Should You Approach Straight Line Problems in JEE Advanced?

1. Identify the Given Constraints First

Before writing any equation, carefully identify all conditions given in the problem. Hidden constraints often determine the correct approach and eliminate unnecessary cases.

2. Draw a Rough Sketch

For every distance, region-based, or intersection problem, make a rough diagram. A simple sketch can save 3–4 minutes of unnecessary algebra and help visualise the geometry.

3. Verify Every Option in Multi-Choice Questions

Do not stop after finding one correct option. Evaluate each option independently because JEE Advanced frequently includes multiple correct answers.

4. Double-Check Sign Conventions

For integer-type and distance-based questions, carefully verify sign conventions in distance and coordinate formulas to avoid calculation errors.

5. Manage Your Time Strategically

If a Straight Line question appears alongside Circles or other Coordinate Geometry topics in Paper 2, consider attempting the Straight Line problem first as a confidence-building question.

Important Exam Insight

The 2014 region-area question is a classic example of how JEE Advanced creatively uses coordinate geometry. The key idea is to divide the first quadrant into separate regions based on whether x or y dominates. Practising at least five similar region-area problems is recommended, as such questions tend to appear approximately once every three years in JEE Advanced.