Question: Two straight lines pass through the origin (๐ฅ', y') = (0,0). One of them passes
through the point (๐ฅ1, ๐ฆ1) = (1,3) and the other passes through the point
(๐ฅ2, ๐ฆ2) = (1,2).
What is the area enclosed between the straight lines in the interval [0, 1] on
the ๐ฅ-axis?
(A) 0.5
(B) 1.0
(C) 1.5
(D) 2.0
(GATE 2022)
Solution:
We can start by finding the ratio of two straight lines through the origin and given points. Let the coordinates of the lines through (0,0) and (1,3) be y = mx and the coordinates of the lines through (0,0) and (1,2) be y = nx. Here the m and n are down to the line.
We can use the formula:
M = (y2 - y1)/(x2 - x1) = (3 - 0)/(1 - 0) = 3/1 =
Therefore, the equation of the first line is y = 3x.
Similarly, we can use the assumption that:
n = (y2 - y1)/(x2 - x1) = (2 - 0)/(1 - 0) = 2/1 =
Therefore, the equation of the second line is y = 2x.
The two lines intersect at the point (0,0) and at the point 2x = 3x, which is x = 0 .
Thus, the area closed between two lines in the interval [0,1] along the x-axis is given by:
∫[0.1] (3x - 2x) dx = ∫[0.1] x dx = [x^2/2] From 0 to 1 = 1/2
Therefore, the answer is (A) 0.5.