Understanding Collinearity of Points

The question asks for the condition under which three specific points, given by their coordinates as \((a, b)\), \((c, d)\), and \((a - c, b - d)\), lie on the same straight line. Points lying on the same straight line are called collinear points. There are several ways to determine if three points are collinear. One common method involves using the concept of the area of a triangle formed by these points.

If three points are collinear, the area of the triangle formed by them is zero. Let's use this property to find the required condition for the given points.

Using the Area of a Triangle Formula for Collinearity

Let the three points be \(A(x_1, y_1) = (a, b)\), \(B(x_2, y_2) = (c, d)\), and \(C(x_3, y_3) = (a - c, b - d)\).

The formula for the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:

Area \(= \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\)

For the points to be collinear, the area must be equal to zero.

Let's substitute the coordinates of points A, B, and C into the area formula:

Area \(= \frac{1}{2} |a(d - (b - d)) + c((b - d) - b) + (a - c)(b - d)|\)

Simplify the terms inside the absolute value:

• First term: \(a(d - b + d) = a(2d - b) = 2ad - ab\)
• Second term: \(c(b - d - b) = c(-d) = -cd\)
• Third term: \((a - c)(b - d) = a(b - d) - c(b - d) = ab - ad - cb + cd\)

Now, sum these terms:

Sum \(= (2ad - ab) + (-cd) + (ab - ad - cb + cd)\)

Combine like terms:

Sum \(= 2ad - ab - cd + ab - ad - cb + cd\)

Sum \(= (2ad - ad) + (-ab + ab) + (-cd + cd) - cb\)

Sum \(= ad + 0 + 0 - cb\)

Sum \(= ad - cb\)

So, the area of the triangle is \(\frac{1}{2} |ad - cb|\).

For the points \((a, b)\), \((c, d)\), and \((a - c, b - d)\) to be collinear, the area must be zero:

\(\frac{1}{2} |ad - cb| = 0\)

This equation holds true if and only if the expression inside the absolute value is zero:

\(ad - cb = 0\)

Rearranging this equation gives the condition for collinearity:

\(ad = cb\)

Comparing with Given Options

Let's compare the derived condition \(ad = cb\) with the given options:

1. \(ab = cd\)
2. \(ac = bd\)
3. \(ad = bc\)
4. \(abc = d\)

The derived condition \(ad = cb\) is the same as \(ad = bc\).

Conclusion on Collinearity Condition

Therefore, the points \((a, b)\), \((c, d)\), and \((a - c, b - d)\) are collinear if and only if the condition \(ad = bc\) is satisfied.

Revision Table: Key Concepts for Collinearity

Additional Information: Exploring Collinearity Further

Besides the area method, another common way to check for collinearity is using slopes. For points \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) to be collinear, the slope of line segment AB must be equal to the slope of line segment BC, provided the denominators are not zero.

Slope of AB = \(\frac{y_2 - y_1}{x_2 - x_1} = \frac{d - b}{c - a}\) (if \(c \neq a\))

Slope of BC = \(\frac{y_3 - y_2}{x_3 - x_2} = \frac{(b - d) - d}{(a - c) - c} = \frac{b - 2d}{a - 2c}\) (if \(a-c \neq c\), i.e., \(a \neq 2c\))

Setting slopes equal:

\(\frac{d - b}{c - a} = \frac{b - 2d}{a - 2c}\)

Cross-multiplying (assuming denominators are non-zero):

\((d - b)(a - 2c) = (b - 2d)(c - a)\)

\(ad - 2cd - ab + 2bc = bc - ab - 2cd + 2ad\)

\(ad - 2cd - ab + 2bc = bc - ab - 2cd + 2ad\)

Notice that \(-2cd\), \(-ab\), and \(2ad\) appear on both sides (with opposite signs for \(2ad\) relative to the original positions but they are on the right side as \(2ad\)). Let's move all terms to one side:

\(ad - 2cd - ab + 2bc - (bc - ab - 2cd + 2ad) = 0\)

\(ad - 2cd - ab + 2bc - bc + ab + 2cd - 2ad = 0\)

Combine like terms:

\((ad - 2ad) + (-2cd + 2cd) + (-ab + ab) + (2bc - bc) = 0\)

\(-ad + 0 + 0 + bc = 0\)

\(-ad + bc = 0\)

\(bc = ad\) or \(ad = bc\).

This confirms the same condition obtained using the area method. The slope method requires careful handling of cases where denominators are zero (vertical or horizontal lines), while the area method handles these cases naturally as the area formula works universally. The condition \(ad = bc\) encompasses all possibilities for the collinearity of these specific points.