NIMCET 2022 — Mathematics PYQ

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Question

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

Area of the parallelogram formed by the lines y= 4x, y =4x + 1, x +y = 0 and x + y = 1 is

Choose the correct answer:

Explanation

A parallelogram formed by two pairs of parallel lines of the form:

$a_{1} x + b_{1} y + c_{1} = 0$

$a_{1} x + b_{1} y + c_{2} = 0$

$a_{2} x + b_{2} y + d_{1} = 0$

$a_{2} x + b_{2} y + d_{2} = 0$

The area is given by the formula:

$Area = \frac{∣ ( c _{1} - c _{2} ) ( d _{1} - d _{2} ) ∣}{∣ a _{1} b _{2} - a _{2} b _{1} ∣}$

1. Rewrite the given equations in standard form ( $a x + b y + c = 0$ ):

Line 1: $4 x - y = 0$
Line 2: $4 x - y + 1 = 0$
Line 3: $x + y = 0$
Line 4: $x + y - 1 = 0$

Here, the coefficients are:

Pair 1: $a_{1} = 4, b_{1} = - 1, c_{1} = 0, c_{2} = 1$
Pair 2: $a_{2} = 1, b_{2} = 1, d_{1} = 0, d_{2} = - 1$

2. Apply the Area Formula:

$Area = \frac{∣ ( 0 - 1 ) ( 0 - ( - 1 )) ∣}{∣ ( 4 ) ( 1 ) - ( 1 ) ( - 1 ) ∣}$

$Area = \frac{∣ ( - 1 ) ( 1 ) ∣}{∣4 + 1∣}$

$Area = \frac{∣ - 1∣}{∣5∣}$

$Area = \frac{1}{5}$

Correct Option: A) $1/5$