If the distance between the points (a, 2) and (3, 4) be 8, then a equals to

Options

A. `2+3sqrt(3)`
B. `2-3 sqrt(15)`
C. `2pm3sqrt(15)`
D. `3pm2sqrt(15)`

Answer

To solve the problem, we need to find the value of \( a \) given that the distance between the points \( (a, 2) \) and \( (3, 4) \) is 8. We will use the distance formula to do this.

### Step-by-Step Solution:

1. **Write the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

2. **Substitute the Given Points**:
Here, the points are \( (a, 2) \) and \( (3, 4) \). Therefore, we can substitute:
\[ d = \sqrt{(3 - a)^2 + (4 - 2)^2} \]

3. **Set the Distance Equal to 8**:
According to the problem, the distance is 8:
\[ \sqrt{(3 - a)^2 + (4 - 2)^2} = 8 \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ (3 - a)^2 + (4 - 2)^2 = 8^2 \]
This simplifies to:
\[ (3 - a)^2 + 2^2 = 64 \]

5. **Calculate \( 2^2 \)**:
We know that \( 2^2 = 4 \), so we can substitute this in:
\[ (3 - a)^2 + 4 = 64 \]

6. **Isolate the Squared Term**:
Subtract 4 from both sides:
\[ (3 - a)^2 = 64 - 4 \]
This simplifies to:
\[ (3 - a)^2 = 60 \]

7. **Take the Square Root**:
Now, we take the square root of both sides:
\[ 3 - a = \pm \sqrt{60} \]

8. **Simplify \( \sqrt{60} \)**:
We can simplify \( \sqrt{60} \):
\[ \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15} \]
Thus, we have:
\[ 3 - a = \pm 2\sqrt{15} \]

9. **Solve for \( a \)**:
Rearranging gives us two equations:
\[ a = 3 - 2\sqrt{15} \quad \text{and} \quad a = 3 + 2\sqrt{15} \]

### Final Answers:
Thus, the values of \( a \) are:
\[ a = 3 - 2\sqrt{15} \quad \text{or} \quad a = 3 + 2\sqrt{15} \]