A line perpendicular to the line segment joining the points $(1, 0)$ and $(2, 3)$ divides it in the ratio $1 : n$ . The equation of the line is

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $3 y + x = \frac{n + 11}{n + 1}$

Detailed Solution

The slope of the segment is $\frac{3 - 0}{2 - 1} = 3$ , so the perpendicular line has slope $- \frac{1}{3}$ .

The point dividing the segment in the ratio $1 : n$ (measured from $(1, 0)$ ) is $(\frac{n \cdot 1 + 1 \cdot 2}{n + 1}, \frac{n \cdot 0 + 1 \cdot 3}{n + 1}) = (\frac{n + 2}{n + 1}, \frac{3}{n + 1})$ .

The line is $y - \frac{3}{n + 1} = - \frac{1}{3} (x - \frac{n + 2}{n + 1})$ , which simplifies to $3 y + x = \frac{9 + n + 2}{n + 1} = \frac{n + 11}{n + 1}$ .

✓ Check

For

n = 1

the point is the midpoint

(\frac{3}{2}, \frac{3}{2})

, and

3 \cdot \frac{3}{2} + \frac{3}{2} = 6 = \frac{12}{2}

⚠ Common trap

Reading the ratio as

n : 1

gives the point

(\frac{2 n + 1}{n + 1}, \frac{3 n}{n + 1})

and the line

3 y + x = \frac{11 n + 1}{n + 1}

, which is not among the options.

Quick tip

Section point $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$ ; perpendicular slope $= - \frac{1}{m}$

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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 perpendicular to the line segment joining the points $(1, 0)$ and $(2, 3)$ divides it in the ratio $1 : n$ . The equation of the line is

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A line perpendicular to the line segment joining the points (1,0) and (2,3) divides it in the ratio 1:n. The equation of the line is

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A line perpendicular to the line segment joining the points $(1, 0)$ and $(2, 3)$ divides it in the ratio $1 : n$ . The equation of the line is