If the points `(a,b),(1,1)and (2,2)` are collinear then

Options

A. `a=2b`
B. `a=3b`
C. `2a=b`
D. `a=b`

Answer

To determine the condition under which the points \((a,b)\), \((1,1)\), and \((2,2)\) are collinear, we can follow these steps:

### Step 1: Understand the concept of collinearity
Three points are collinear if the slope between any two pairs of points is the same.

### Step 2: Calculate the slope between the points \((1,1)\) and \((2,2)\)
Using the slope formula:
\[ M_2 = \frac{y_2 - y_1}{x_2 - x_1} \]
For the points \((1,1)\) and \((2,2)\):
\[ M_2 = \frac{2 - 1}{2 - 1} = \frac{1}{1} = 1 \]

### Step 3: Calculate the slope between the points \((a,b)\) and \((1,1)\)
Using the slope formula again:
\[ M_1 = \frac{1 - b}{1 - a} \]

### Step 4: Set the slopes equal to each other
Since the points are collinear, we can set \(M_1\) equal to \(M_2\):
\[ \frac{1 - b}{1 - a} = 1 \]

### Step 5: Solve for \(a\) and \(b\)
Cross-multiply to eliminate the fraction:
\[ 1 - b = 1 - a \]
Now, simplify the equation:
\[ -b = -a \]
This leads to:
\[ a = b \]

### Conclusion
Thus, the condition for the points \((a,b)\), \((1,1)\), and \((2,2)\) to be collinear is:
\[ a = b \]