Two straight lines pass through the origin (𝑥', y') = (0,0). One of them passes through the point (𝑥1, 𝑦1) = (1,3) and the other passes through the point (𝑥2, 𝑦2) = (1,2). What is the area enclosed between the straight lines in the interval [0, 1] on the 𝑥-axis? | GATE 2022

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

through the point (𝑥1, 𝑦1) = (1,3) and the other passes through the point

(𝑥2, 𝑦2) = (1,2).

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

the 𝑥-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.