In a ` A B C ,A-=(alpha,beta),B-=(1,2),C-=(2,3),` point `A` lies on the line `y=2x+3,` where `alpha,beta` are integers, and the area of the triangle is `S` such that `[S]=2` where `[` .`]` denotes the greatest integer function. Then the possible coordinates of `A` can be `(-7,-11)` `(-6,-9)` `(2,7)` `(3,9)`

Options

A. `(-7,-11)`
B. `(-6,-9)`
C. (2,7)
D. (3,9)

Explanation

To solve the problem step by step, we will find the coordinates of point A (α, β) such that it lies on the line \(y = 2x + 3\) and the area of triangle ABC is such that \([S] = 2\), where \([S]\) denotes the greatest integer function.

### Step 1: Determine the coordinates of point A
Since point A lies on the line \(y = 2x + 3\), we can express the coordinates of point A as:
\[ A = (α, β) = (α, 2α + 3) \]

### Step 2: Identify the coordinates of points B and C
The coordinates of points B and C are given as:
\[ B = (1, 2), \quad C = (2, 3) \]

### Step 3: Use the formula for the area of a triangle
The area \(S\) of triangle ABC can be calculated using the formula:
\[ S = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points A, B, and C into this formula:
\[ S = \frac{1}{2} \left| α(2 - 3) + 1(3 - (2α + 3)) + 2((2α + 3) - 2) \right| \]

### Step 4: Simplify the area expression
Substituting and simplifying:
\[ S = \frac{1}{2} \left| α(-1) + 1(3 - 2α - 3) + 2(2α + 3 - 2) \right| \]
\[ = \frac{1}{2} \left| -α + (-2α) + 2(2α + 1) \right| \]
\[ = \frac{1}{2} \left| -α - 2α + 4α + 2 \right| \]
\[ = \frac{1}{2} \left| α + 2 \right| \]

### Step 5: Set the area condition
We are given that \([S] = 2\). This means:
\[ 2 \leq S < 3 \]
Substituting for \(S\):
\[ 2 \leq \frac{1}{2} |α + 2| < 3 \]

### Step 6: Solve the inequalities
Multiplying through by 2:
\[ 4 \leq |α + 2| < 6 \]

This gives us two cases to consider for \( |α + 2| \):
1. \(α + 2 \geq 4\) or \(α + 2 \leq -4\)
2. \(α + 2 < 6\) and \(α + 2 > -6\)

From \(α + 2 \geq 4\):
\[ α \geq 2 \]

From \(α + 2 < 6\):
\[ α < 4 \]

From \(α + 2 \leq -4\):
\[ α \leq -6 \]

From \(α + 2 > -6\):
\[ α > -8 \]

### Step 7: Combine the results
Combining these inequalities, we have two ranges:
1. \(2 \leq α < 4\)
2. \(-8 < α \leq -6\)

### Step 8: Determine integer values for α
The integer values of \(α\) that satisfy these conditions are:
- From \(2 \leq α < 4\): \(α = 2, 3\)
- From \(-8 < α \leq -6\): \(α = -7, -6\)

### Step 9: Calculate corresponding β values
Now, we can find the corresponding \(β\) values using \(β = 2α + 3\):
1. For \(α = 2\): \(β = 2(2) + 3 = 7\) → Point A = (2, 7)
2. For \(α = 3\): \(β = 2(3) + 3 = 9\) → Point A = (3, 9)
3. For \(α = -7\): \(β = 2(-7) + 3 = -11\) → Point A = (-7, -11)
4. For \(α = -6\): \(β = 2(-6) + 3 = -9\) → Point A = (-6, -9)

### Final Possible Coordinates of A
The possible coordinates of point A are:
- \((-7, -11)\)
- \((-6, -9)\)
- \((2, 7)\)
- \((3, 9)\)