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

Solution:

According to the section formula, the coordinates of the point that divides the line segment joining the points (1, 0) and (2, 3) in the ratio 1: n is given by

[n (1) + 1(2)/(1 + n), n (0) + 1(3)/(1 + n)] = [(n + 2)/(n + 1), 3//(n + 1)]

The slope of the line joining the points (1, 0) and (2, 3) is m = (3 - 0)/(2 - 1) = 3

We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.

Therefore, slope of the line that is perpendicular to the line joining the points (1, 0) and (2, 3) is - 1/m = - 1/3

Now the equation of the line passing through [(n + 2)/(n + 1), 3//(n + 1)] and whose slope is - 1/3, given by

[y - 3//(n + 1)] = - 1/3 [x - (n + 2)/(n + 1)]

3 [(n + 1) y - 3] = - [x (n + 1) - (n - 2)]

3(n + 1) y - 9 = -(n + 1) x + n + 2

(1 + n) x + 3(1 + n) y = n + 11

Hence, the equation of the line is (1 + n) x + 3(1 + n) y = n + 11

NCERT Solutions Class 11 Maths Chapter 10 Exercise 10.2 Question 11

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

Summary:

The equation of the line.perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1: n is (1 + n) x + 3(1 + n) y = n + 11