The points (-a, -b), (0, 0), (a, b) and (a2, ab) are:1. Vertices of a

1.	The points (-a, -b), (0, 0), (a, b) and (a2, ab) are:1. Vertices of a parallelogram.2. Collinear.3. Vertices of a rectangle.4. None of these.
Answer» Correct Answer - Option 2 : Collinear. Concept: The slope m of the line joining the two points (x₁, y₁) and (x₂, y₂) is m = \(\rm \frac{y_1-y_2}{x_1-x_2}\). The slopes of parallel lines are the same. If the lines have the same slope and also have a common point, then they are the same line. Calculation: Let's say that the points are A(-a, -b), B(0, 0), C(a, b) and D(a², ab). Using the slope formula, the slope of the line AB will be: m_AB = \(\rm \frac{-b-0}{-a-0}=\frac{-b}{-a}=\frac{b}{a}\) Also, the slopes of the lines BC, CD and AD are: m_BC = \(\rm \frac{0-b}{0-a}=\frac{-b}{-a}=\frac{b}{a}\) m_CD = \(\rm \frac{b-ab}{a-{a}^{2}}=\frac{b(1-a)}{a(1-a)}=\frac{b}{a}\) m_AD = \(\rm \frac{-b-ab}{-a-{a}^{2}}=\frac{b(-1-a)}{a(-1-a)}=\frac{b}{a}\) Since the slopes of all the lines are same and they also have some common points between them, we can say that the points are actually collinear. The correct answer is B. Collinear.

The points (-a, -b), (0, 0), (a, b) and (a2, ab) are:1. Vertices of a parallelogram.2. Collinear.3. Vertices of a rectangle.4. None of these.

Answer» Correct Answer - Option 2 : Collinear.

Concept:

The slope m of the line joining the two points (x₁, y₁) and (x₂, y₂) is m = \(\rm \frac{y_1-y_2}{x_1-x_2}\).
The slopes of parallel lines are the same.
If the lines have the same slope and also have a common point, then they are the same line.

Calculation:

Let's say that the points are A(-a, -b), B(0, 0), C(a, b) and D(a², ab).

Using the slope formula, the slope of the line AB will be:

m_AB = \(\rm \frac{-b-0}{-a-0}=\frac{-b}{-a}=\frac{b}{a}\)

Also, the slopes of the lines BC, CD and AD are:

m_BC = \(\rm \frac{0-b}{0-a}=\frac{-b}{-a}=\frac{b}{a}\)

m_CD = \(\rm \frac{b-ab}{a-{a}^{2}}=\frac{b(1-a)}{a(1-a)}=\frac{b}{a}\)

m_AD = \(\rm \frac{-b-ab}{-a-{a}^{2}}=\frac{b(-1-a)}{a(-1-a)}=\frac{b}{a}\)

Since the slopes of all the lines are same and they also have some common points between them, we can say that the points are actually collinear.

The correct answer is B. Collinear.