Area of the parallelogram formed by the lines y = mx, y = mx + 1, y = nx and y = nx + 1 equals
(a) |

Q: Why can't we just find the vertices and use the determinant formula? Isn't that safer?

You absolutely can, and in a high-pressure exam, if you forget the shortcut, that is exactly what you should do! However, the determinant (or shoelace) method is computationally expensive and prone to arithmetic errors. The shortcut formula is a derived result that saves you precious minutes. In JEE Advanced, time is currency. Mastering these shortcuts is how you move from 'solving' to 'dominating' the paper.

Q: What if the lines were vertical? Does the formula break?

That is a brilliant observation! Yes, the slope-intercept form $y = mx + c$ fails for vertical lines because the slope is undefined. If you encounter lines like $x = k_1$ and $x = k_2$, you would need to switch to the $x = ky + c$ form or use a different geometric approach. Always check the domain of your tools before applying them!

Q: Why is the denominator $|m - n|$? Why not $|m + n|$?

Think about the angle between the lines. The area of a parallelogram is given by $\text{Base} \times \text{Height}$. The height depends on the sine of the angle between the intersecting lines. The tangent of the angle between two lines with slopes $m$ and $n$ is $|\frac{m - n}{1 + mn}|$. As the slopes $m$ and $n$ get closer to each other, the angle between them shrinks, and the area grows larger. The term $|m - n|$ in the denominator captures this inverse relationship perfectly.

Question

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

Options:

(A) $|m + n|/(m - n)^2$

(B) $2/|m + n|$

(C) $1/|m + n|$

(D) $1/|m - n|$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, my dear student. Today, we are not just solving a problem; we are peeling back the layers of coordinate geometry to reveal the elegance hidden beneath the algebra. When you look at the equations $y = mx$, $y = mx + 1$, $y = nx$, and $y = nx + 1$, do not see a jumble of variables. Train your eyes to see a structured grid.

### The Parallel Tracks

Imagine you are standing on a vast, infinite plane. You draw a line, $y = mx$, which passes through the origin. Now, you draw another line, $y = mx + 1$. Because they share the same slope $m$, they are like two parallel train tracks that will never meet.

The distance between them is fixed, determined by that constant shift of $1$. This is our first pair of parallel lines. We then introduce the second pair: $y = nx$ and $y = nx + 1$. These share the slope $n$ and, when they cross the first set, they create a beautiful, enclosed, four-sided shape: a **parallelogram**.

### The Power of the Shortcut

The standard, brute-force approach involves finding the four intersection points and applying the shoelace formula. While effective, it is the path of the novice. We are aiming for the path of the master.

There is a powerful, derived formula for the area of a parallelogram formed by two pairs of parallel lines: $y = m_1x + c_1$, $y = m_1x + c_2$, $y = m_2x + d_1$, and $y = m_2x + d_2$. The area is given by:

$$	ext{Area} = \frac{|(c_1 - c_2)(d_1 - d_2)|}{|m_1 - m_2|}$$

Think of the area as the product of the distances between the parallel lines, divided by the sine of the angle between them. When you perform the coordinate geometry derivation, all the trigonometric complexity collapses into this simple, elegant ratio.

### The Execution

Let us map our problem to this formula. Our first pair is $y = mx + 0$ and $y = mx + 1$, where $m_1 = m$, $c_1 = 0$, and $c_2 = 1$. Our second pair is $y = nx + 0$ and $y = nx + 1$, where $m_2 = n$, $d_1 = 0$, and $d_2 = 1$.

Plugging these values into our formula:

$$	ext{Area} = \frac{|(0 - 1)(0 - 1)|}{|m - n|}$$

The numerator becomes $|(-1)(-1)|$, which is $1$. The denominator is the absolute difference of the slopes, $|m - n|$. Thus, our final area is:

$$	ext{Area} = \frac{1}{|m - n|}$$

### The Intuition Check

Before we conclude, let us verify if this result makes physical sense. Look at the denominator, $|m - n|$. If $m$ is very close to $n$, the lines are almost parallel and will intersect very far away, creating a massive, elongated parallelogram.

As $|m - n| 	o 0$, the area approaches infinity, which matches our formula perfectly. Conversely, if $|m - n|$ is very large, the lines are nearly perpendicular, creating a compact, smaller area. The formula is not just a collection of symbols; it is a reflection of the geometric reality of the plane.

Animated Solution for Mathematics - Straight Lines: Area of the parallelogram formed by the lines $y = m x, y = m x + 1, y = n x$ and $y = n x + 1$ equals

Select Answer:

Visualized Solution

Visualizing the First Pair of Parallel Lines

Adding the Second Pair of Parallel Lines

Identifying the Enclosed Parallelogram

The Standard Area Formula Tool

Understanding the Formula's Origin

Mapping Our Equations to the Formula

Calculating the Numerator

Assembling the Final Area Expression

Final Takeaway and Intuition Check

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

The Parallel Tracks

The Power of the Shortcut

The Execution

The Intuition Check

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Animated Solution for Mathematics - Straight Lines: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Area of the parallelogram formed by the lines y=mx,y=mx+1,y=nx and y=nx+1 equals