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 step by step, we will use the distance formula and the information provided in the question.

### Step 1: Understand the distance formula
The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 2: Identify the points
In this case, the points are \((a, 2)\) and \((3, 4)\). Here, \(x_1 = a\), \(y_1 = 2\), \(x_2 = 3\), and \(y_2 = 4\).

### Step 3: Set up the equation
We are given that the distance between these two points is 8. Therefore, we can set up the equation:
\[ \sqrt{(3 - a)^2 + (4 - 2)^2} = 8 \]

### Step 4: Simplify the equation
First, simplify \( (4 - 2)^2 \):
\[ (4 - 2)^2 = 2^2 = 4 \]
Now, substitute this back into the equation:
\[ \sqrt{(3 - a)^2 + 4} = 8 \]

### Step 5: Square both sides
To eliminate the square root, square both sides of the equation:
\[ (3 - a)^2 + 4 = 8^2 \]
This simplifies to:
\[ (3 - a)^2 + 4 = 64 \]

### Step 6: Isolate the squared term
Subtract 4 from both sides:
\[ (3 - a)^2 = 64 - 4 \]
\[ (3 - a)^2 = 60 \]

### Step 7: Take the square root
Now, take the square root of both sides:
\[ 3 - a = \pm \sqrt{60} \]

### Step 8: Solve for \( a \)
This gives us two equations:
1. \( 3 - a = \sqrt{60} \)
2. \( 3 - a = -\sqrt{60} \)

For the first equation:
\[ a = 3 - \sqrt{60} \]

For the second equation:
\[ a = 3 + \sqrt{60} \]

### Step 9: Simplify \( \sqrt{60} \)
We can simplify \( \sqrt{60} \):
\[ \sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15} \]
Thus, we have:
1. \( a = 3 - 2\sqrt{15} \)
2. \( a = 3 + 2\sqrt{15} \)

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