A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts $p$ and $q$ . Which of the following is true?

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — $\frac{1}{a ^{2}} + \frac{1}{b ^{2}} = \frac{1}{p ^{2}} + \frac{1}{q ^{2}}$

Detailed Solution

The line $\frac{x}{a} + \frac{y}{b} = 1$ is at distance $d$ from the origin, where $\frac{1}{d ^{2}} = \frac{1}{a ^{2}} + \frac{1}{b ^{2}}$ .

Rotating the axes about the origin does not change this distance; in the new axes the line is $\frac{X}{p} + \frac{Y}{q} = 1$ , so $\frac{1}{d ^{2}} = \frac{1}{p ^{2}} + \frac{1}{q ^{2}}$ .

Hence $\frac{1}{a ^{2}} + \frac{1}{b ^{2}} = \frac{1}{p ^{2}} + \frac{1}{q ^{2}}$ .

✓ Check

Rotating by

90^{\circ}

turns the intercepts into

p = b

q = - a

, and

\frac{1}{b ^{2}} + \frac{1}{a ^{2}}

is unchanged.

⚠ Common trap

Option (A) assumes the intercept segment keeps its length; for

a = 2

b = 3

and a

45^{\circ}

rotation,

p^{2} + q^{2} \approx 75 \neq = 13

Quick tip

Distance from origin is invariant: $\frac{1}{a ^{2}} + \frac{1}{b ^{2}} = \frac{1}{p ^{2}} + \frac{1}{q ^{2}}$

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Straight Line — Preparation

This question tests your understanding of Straight Line, 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 straight line is the most fundamental topic in coordinate geometry. It covers the slope of a line, the angle between two lines, conditions for parallel and perpendicular lines, and the different forms of the equation of a line: slope-intercept, point-slope, two-point, intercept and normal forms. Exam questions also test the perpendicular distance of a point from a line, the distance between two parallel lines, the point of intersection of two lines, concurrency of three lines, and families of lines passing through an intersection point. Remember that two lines are perpendicular when the product of their slopes is -1 and parallel when their slopes are equal. Picking the right form of the equation for the information given is the main skill. Practise the straight line questions below, each with a step-by-step solution.

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A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts $p$ and $q$ . Which of the following is true?

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Straight Line — Preparation

A line L has intercepts a and b on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts p and q. Which of the following is true?

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Straight Line — Preparation

A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line has intercepts $p$ and $q$ . Which of the following is true?