The value (s) of x for which the area of triangle formed by the points `(5,-1),(x,-3)and (6,3)is 11/2` unit square is/are

Options

A. `15/2`
B. 2
C. Both 1 & 2
D. neither 1 nor 2

Answer

To find the value(s) of \( x \) for which the area of the triangle formed by the points \( (5, -1) \), \( (x, -3) \), and \( (6, 3) \) is \( \frac{11}{2} \) square units, we can use the formula for the area of a triangle given by its vertices.

### Step-by-Step Solution:

1. **Identify the Points:**
Let the points be:
- \( A(5, -1) \)
- \( B(x, -3) \)
- \( C(6, 3) \)

2. **Area Formula:**
The area \( A \) of a triangle formed by points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

3. **Substituting the Points:**
Substituting the coordinates of points \( A \), \( B \), and \( C \):
\[ A = \frac{1}{2} \left| 5(-3 - 3) + x(3 - (-1)) + 6(-1 - (-3)) \right| \]
Simplifying this:
\[ A = \frac{1}{2} \left| 5(-6) + x(4) + 6(2) \right| \]
\[ A = \frac{1}{2} \left| -30 + 4x + 12 \right| \]
\[ A = \frac{1}{2} \left| 4x - 18 \right| \]

4. **Setting the Area Equal to \( \frac{11}{2} \):**
We set the area equal to \( \frac{11}{2} \):
\[ \frac{1}{2} \left| 4x - 18 \right| = \frac{11}{2} \]
Multiplying both sides by 2:
\[ \left| 4x - 18 \right| = 11 \]

5. **Solving the Absolute Value Equation:**
This gives us two cases to solve:
- Case 1: \( 4x - 18 = 11 \)
- Case 2: \( 4x - 18 = -11 \)

**Case 1:**
\[ 4x - 18 = 11 \]
\[ 4x = 29 \]
\[ x = \frac{29}{4} \]

**Case 2:**
\[ 4x - 18 = -11 \]
\[ 4x = 7 \]
\[ x = \frac{7}{4} \]

6. **Final Values of \( x \):**
The values of \( x \) for which the area of the triangle is \( \frac{11}{2} \) square units are:
\[ x = \frac{29}{4} \quad \text{and} \quad x = \frac{7}{4} \]