If the point (1,2) and (34) were to be on the same side of the line `3x-5y+a=0` then

Options

A. `7 lt a lt 11`
B. a=7
C. a=11
D. `a lt 7 or a gt 11`

Answer

To determine the range of values of \( a \) such that the points \( (1, 2) \) and \( (3, 4) \) are on the same side of the line given by the equation \( 3x - 5y + a = 0 \), we can follow these steps:

### Step 1: Define the line equation and points
The line equation is given as:
\[ 3x - 5y + a = 0 \]
The points are \( P(1, 2) \) and \( R(3, 4) \).

### Step 2: Substitute the points into the line equation
We will substitute the coordinates of the points into the line equation to find expressions for each point.

For point \( P(1, 2) \):
\[ L_1 = 3(1) - 5(2) + a = 3 - 10 + a = a - 7 \]

For point \( R(3, 4) \):
\[ L_2 = 3(3) - 5(4) + a = 9 - 20 + a = a - 11 \]

### Step 3: Determine the condition for the points to be on the same side
The points \( P \) and \( R \) will be on the same side of the line if the product of \( L_1 \) and \( L_2 \) is greater than 0:
\[ L_1 \cdot L_2 > 0 \]
This means:
\[ (a - 7)(a - 11) > 0 \]

### Step 4: Solve the inequality
To solve the inequality \( (a - 7)(a - 11) > 0 \), we can find the critical points by setting each factor to zero:
\[ a - 7 = 0 \quad \Rightarrow \quad a = 7 \]
\[ a - 11 = 0 \quad \Rightarrow \quad a = 11 \]

### Step 5: Analyze the intervals
The critical points divide the number line into three intervals:
1. \( (-\infty, 7) \)
2. \( (7, 11) \)
3. \( (11, \infty) \)

We will test a point from each interval to determine where the product is positive.

- For \( a < 7 \) (e.g., \( a = 0 \)):
\[ (0 - 7)(0 - 11) = (-7)(-11) = 77 > 0 \quad \text{(True)} \]

- For \( 7 < a < 11 \) (e.g., \( a = 9 \)):
\[ (9 - 7)(9 - 11) = (2)(-2) = -4 < 0 \quad \text{(False)} \]

- For \( a > 11 \) (e.g., \( a = 12 \)):
\[ (12 - 7)(12 - 11) = (5)(1) = 5 > 0 \quad \text{(True)} \]

### Step 6: Conclusion
The points \( (1, 2) \) and \( (3, 4) \) are on the same side of the line \( 3x - 5y + a = 0 \) when:
\[ a < 7 \quad \text{or} \quad a > 11 \]

### Final Answer
The range of values for \( a \) is:
\[ (-\infty, 7) \cup (11, \infty) \]