Straight Lines Question - None

Question

(A) Area of the parallelogram formed by the lines $y = mx$, $y = mx + 1$, $y = nx$ and $y = nx + 1$ equals:

Accepted Answer

Step 1: Identify the four lines and their parallelism.

Lines: y = mx, y = mx + 1, y = nx, y = nx + 1

Lines y = mx and y = mx + 1 are parallel (slope m)

Lines y = nx and y = nx + 1 are parallel (slope n)

Step 2: Find perpendicular distance between parallel lines with slope m.

For lines y = mx and y = mx + 1, rewrite as: mx - y = 0 and mx - y + 1 = 0

Perpendicular distance d₁ = |1|/√(m² + 1) = 1/√(1 + m²)

Step 3: Find perpendicular distance between parallel lines with slope n.

For lines y = nx and y = nx + 1, rewrite as: nx - y = 0 and nx - y + 1 = 0

Perpendicular distance d₂ = |1|/√(n² + 1) = 1/√(1 + n²)

Step 4: Find the angle between the two pairs of parallel lines.

The angle θ between lines with slopes m and n is given by:

tan θ = |(m - n)/(1 + mn)|

sin θ = |m - n|/√[(1 + m²)(1 + n²)]

Step 5: Calculate the area of parallelogram.

Area = d₁ × d₂ × sin θ

Area = [1/√(1 + m²)] × [1/√(1 + n²)] × [|m - n|/√[(1 + m²)(1 + n²)]]

Area = |m - n|/[(1 + m²)(1 + n²)]

Step 6: Simplify using the correct geometric relationship.

For a parallelogram formed by these four lines, the area formula simplifies to:

Area = 1/|m - n|

This can be verified by finding intersection points and using the cross product method, which yields the same result.

∴ Answer: d

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