If the point P(2, 2) is equidistant from the points A(-2, k) and B(-2k, -3), find k. Also, find the length of AP.

Explanation

To solve the problem, we need to find the value of \( k \) such that the point \( P(2, 2) \) is equidistant from the points \( A(-2, k) \) and \( B(-2k, -3) \). We will also find the length of \( AP \).

### Step 1: Set up the distance equations

Since \( P \) is equidistant from \( A \) and \( B \), we can set the distances equal:

\[ AP = PB \]

The distance \( AP \) can be calculated using the distance formula:

\[ AP = \sqrt{(2 - (-2))^2 + (2 - k)^2} = \sqrt{(2 + 2)^2 + (2 - k)^2} = \sqrt{4^2 + (2 - k)^2} \]

The distance \( PB \) is:

\[ PB = \sqrt{(2 - (-2k))^2 + (2 - (-3))^2} = \sqrt{(2 + 2k)^2 + (2 + 3)^2} = \sqrt{(2 + 2k)^2 + 5^2} \]

### Step 2: Square both sides to eliminate the square roots

Squaring both distances gives:

\[ (4^2 + (2 - k)^2) = ((2 + 2k)^2 + 5^2) \]

This simplifies to:

\[ 16 + (2 - k)^2 = (2 + 2k)^2 + 25 \]

### Step 3: Expand both sides

Expanding the left side:

\[ 16 + (2 - k)^2 = 16 + (4 - 4k + k^2) = 16 + 4 - 4k + k^2 = 20 - 4k + k^2 \]

Expanding the right side:

\[ (2 + 2k)^2 + 25 = (4 + 8k + 4k^2) + 25 = 4k^2 + 8k + 29 \]

### Step 4: Set the equations equal

Now we have:

\[ 20 - 4k + k^2 = 4k^2 + 8k + 29 \]

### Step 5: Rearrange the equation

Rearranging gives:

\[ 0 = 4k^2 + 8k + 29 - 20 + 4k - k^2 \]

Combining like terms:

\[ 0 = 3k^2 + 12k + 9 \]

### Step 6: Factor the quadratic equation

Factoring out a common factor of 3:

\[ 0 = k^2 + 4k + 3 \]

Factoring further:

\[ 0 = (k + 1)(k + 3) \]

### Step 7: Solve for \( k \)

Setting each factor to zero gives:

\[ k + 1 = 0 \quad \Rightarrow \quad k = -1 \]
\[ k + 3 = 0 \quad \Rightarrow \quad k = -3 \]

### Step 8: Find the length of \( AP \)

Now we will find the length of \( AP \) for both values of \( k \).

1. For \( k = -1 \):
\[ A(-2, -1) \Rightarrow AP = \sqrt{(2 - (-2))^2 + (2 - (-1))^2} = \sqrt{(2 + 2)^2 + (2 + 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

2. For \( k = -3 \):
\[ A(-2, -3) \Rightarrow AP = \sqrt{(2 - (-2))^2 + (2 - (-3))^2} = \sqrt{(2 + 2)^2 + (2 + 3)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]

### Final Answers:

- The values of \( k \) are \( -1 \) and \( -3 \).
- The lengths of \( AP \) are \( 5 \) and \( \sqrt{41} \).