Find the value of 'a' if the distance between the points (a,2), (3,4) is `2sqrt2.`

Answer

To find the value of 'a' such that the distance between the points (a, 2) and (3, 4) is \(2\sqrt{2}\), we will use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 1: Identify the points
Let the points be:
- Point 1: \((x_1, y_1) = (a, 2)\)
- Point 2: \((x_2, y_2) = (3, 4)\)

### Step 2: Apply the distance formula
Using the distance formula, we can express the distance between the two points as:

\[ d = \sqrt{(3 - a)^2 + (4 - 2)^2} \]

### Step 3: Simplify the equation
We know that the distance \(d\) is given as \(2\sqrt{2}\). Thus, we can set up the equation:

\[ \sqrt{(3 - a)^2 + (4 - 2)^2} = 2\sqrt{2} \]

Calculating \(4 - 2\):

\[ 4 - 2 = 2 \implies (4 - 2)^2 = 2^2 = 4 \]

So, the equation becomes:

\[ \sqrt{(3 - a)^2 + 4} = 2\sqrt{2} \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:

\[ (3 - a)^2 + 4 = (2\sqrt{2})^2 \]

Calculating \((2\sqrt{2})^2\):

\[ (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]

Thus, we have:

\[ (3 - a)^2 + 4 = 8 \]

### Step 5: Rearrange the equation
Subtract 4 from both sides:

\[ (3 - a)^2 = 8 - 4 \]
\[ (3 - a)^2 = 4 \]

### Step 6: Take the square root
Taking the square root of both sides gives us two cases:

\[ 3 - a = 2 \quad \text{or} \quad 3 - a = -2 \]

### Step 7: Solve for 'a'
**Case 1:**

\[ 3 - a = 2 \implies a = 3 - 2 = 1 \]

**Case 2:**

\[ 3 - a = -2 \implies a = 3 + 2 = 5 \]

### Step 8: Final values
Thus, the values of \(a\) are:

\[ a = 1 \quad \text{or} \quad a = 5 \]

### Summary
The values of \(a\) that satisfy the given condition are \(1\) and \(5\).

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