The points (-a, -b), (0, 0), (a, b) and (a², ab) are:
(a) Collinear
(b) Vertices of a parallelogram

Q: Wait, what if $a = 0$? Doesn't the slope formula fail then?

That is a brilliant observation! If $a = 0$, the points become $(0, -b), (0, 0), (0, b),$ and $(0, 0)$. They all lie on the y-axis. In this specific case, the slope is undefined, but the points are still collinear because they all lie on the line $x = 0$. The logic holds!

Q: Why did we use slopes instead of the distance formula?

You could use the distance formula, but it would be much more tedious! Calculating the distance between all pairs to check if $AB + BC = AC$ is a lot of work. Slopes are elegant because they tell you the 'direction' of the points instantly, making the verification process much faster.

Q: Does the order of points matter for collinearity?

Not at all! Collinearity is a property of the set of points. As long as they all satisfy the same linear equation, they are collinear, regardless of their order on the line. You are just checking if they all 'belong' to the same straight path.

Question

The points $(-a, -b), (0, 0), (a, b)$ and $(a^2, ab)$ are:

Options:

(A) Collinear

(B) Vertices of a parallelogram

(C) Vertices of a rectangle

(D) None of these

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a vast, empty coordinate plane. You have four points scattered before you: $A(-a, -b)$, $B(0, 0)$, $C(a, b)$, and $D(a^2, ab)$.

At first glance, they look like a random collection of coordinates, but there is a hidden harmony here. In coordinate geometry, when we ask if points are collinear, we are really asking a deeper question: Do these points share a single, unified path?

Are they all marching to the beat of the same linear drum?

### The Power of Slope

To uncover this truth, we need a tool that measures the 'steepness' or 'direction' of a path. That tool is the slope, defined as:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Think of the slope as the DNA of a line. If a set of points is collinear, the slope between any two of them must be identical.

It is like checking the angle of a staircase; if the angle changes, you are on a different staircase. If it stays the same, you are on the same flight.

### The Calculation Journey

Let us embark on this calculation. First, we look at the segment $AB$, connecting $A(-a, -b)$ and $B(0, 0)$.

Applying our formula, we get:

$$m_{AB} = \frac{0 - (-b)}{0 - (-a)} = \frac{b}{a}$$

Now, let us check the segment $BC$, connecting $B(0, 0)$ and $C(a, b)$. The slope is:

$$m_{BC} = \frac{b - 0}{a - 0} = \frac{b}{a}$$

The consistency is already emerging! Finally, we face the most intimidating point, $D(a^2, ab)$. We calculate the slope of $BD$ as:

$$m_{BD} = \frac{ab - 0}{a^2 - 0}$$

This looks complex, but watch what happens when we simplify. By canceling the common factor $a$ (assuming $a 
eq 0$), we are left with:

$$m_{BD} = \frac{ab}{a^2} = \frac{b}{a}$$

### The Aha! Moment

Look at that! We have established that:

$$m_{AB} = m_{BC} = m_{BD} = \frac{b}{a}$$

Every single segment we tested yields the exact same slope. This is not a coincidence; it is the geometric reality of these points.

Because they share a common point $B$ and maintain the same slope, they are locked into the same straight line. We have successfully proven that these points are **collinear**.

This problem teaches us that even when coordinates look like abstract variables, they often obey simple, elegant rules. Keep looking for these patterns, and you will find that mathematics is not just about solving equations—it is about uncovering the beautiful, hidden order of the universe.

Animated Solution for Mathematics - Straight Lines: The points $(- a, - b), (0, 0), (a, b)$ and $(a^{2}, ab)$ are:

Select Answer:

Visualized Solution

Visualizing the Given Points

The Condition for Collinearity

Setting up Slope for Segment $A B$

Simplifying Slope $m_{A B}$

Slope of Segment $B C$

Setting up Slope for Segment $B D$

Simplifying Slope $m_{B D}$

Final Conclusion: Collinear Points

The Sigma Insight: Slope of a Line

Analyzing the Setup

The Power of Slope

The Calculation Journey

The Aha! Moment

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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; } The points (−a,−b),(0,0),(a,b) and (a2,ab) are: