# Two straight lines pass through the origin (๐ฅ', y') = (0,0). One of them passes through the point (๐ฅ1, ๐ฆ1) = (1,3) and the other passes through the point (๐ฅ2, ๐ฆ2) = (1,2). What is the area enclosed between the straight lines in the interval [0, 1] on the ๐ฅ-axis? | GATE 2022

Two straight lines pass through the origin (๐ฅ', y') = (0,0). One of them passes through the point (๐ฅ1, ๐ฆ1) = (1,3) and the other passes through th

Question: Two straight lines pass through the origin (๐ฅ', y') = (0,0). One of them passes

through the point (๐ฅ1, ๐ฆ1) = (1,3) and the other passes through the point

(๐ฅ2, ๐ฆ2) = (1,2).

What is the area enclosed between the straight lines in the interval [0, 1] on

the ๐ฅ-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.

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