Two straight lines pass through the origin (x_0, y_0) = (0,0). One of them passes through the point (x_1, y_1) = (1,3) and the other passes through the point (x_2, y_2) = (1,2). What is the area enclosed between the straight lines in the interval [0,1] on the x-axis?

Previously Asked in : Gate CE-forenoon season 2022 ||

0.5

1.0

1.5

2.0

Detailed Explanation :

Equation of first straight line passing through (0,0) and (1,3)

\Rightarrow y - y_1 =\left ( \dfrac {y_2 - y_1} {x_2 - x_1} \right) (x - x_1 )

\Rightarrow y - 0 =\left ( \dfrac {3-0} {1-0} \right) ( x - 0)

y = 3x

Equation of second straight line passing through (0,0) and (1,2)

\Rightarrow y - y_1 = \left( \dfrac {y_2 - y_1} {x_2 - x_1} \right) (x - x_1)

\Rightarrow y - 0 = \left( \dfrac {2 - 0} {1 - 0} \right) (x - 0)

y = 2x

Area = \int_{0}^{1} (3x - 2x) \,dx)

= \bigg( \frac {3x^2} {2} - x^2 \bigg)_{0}^{1}

= \dfrac {1} {2} = 0.5

Another shortcut way is :

Area ABC, =Area (ABD) – Area (ACD)

= \frac {1}{2} \times 3 \times 1 - \frac {1} {2} \times 2 \times 1 = \frac {1} {2}

= 0.5 {unit}^2

Correct option is 0.5

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