Find the locus of a point equidistant from the point (2,4) and the y-axis.

Answer

To find the locus of a point equidistant from the point (2, 4) and the y-axis, we can follow these steps:

### Step 1: Define the point on the locus
Let \( P(h, k) \) be any point on the locus.

### Step 2: Calculate the distance from point \( P \) to the point (2, 4)
Using the distance formula, the distance \( d_1 \) from point \( P(h, k) \) to the point (2, 4) is given by:
\[ d_1 = \sqrt{(h - 2)^2 + (k - 4)^2} \]

### Step 3: Calculate the distance from point \( P \) to the y-axis
The distance \( d_2 \) from point \( P(h, k) \) to the y-axis is simply the absolute value of the x-coordinate, which is:
\[ d_2 = |h| \]

### Step 4: Set the distances equal
Since the point \( P \) is equidistant from (2, 4) and the y-axis, we set the distances equal:
\[ \sqrt{(h - 2)^2 + (k - 4)^2} = |h| \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (h - 2)^2 + (k - 4)^2 = h^2 \]

### Step 6: Expand both sides
Expanding the left side:
\[ (h^2 - 4h + 4) + (k^2 - 8k + 16) = h^2 \]
This simplifies to:
\[ h^2 - 4h + 4 + k^2 - 8k + 16 = h^2 \]

### Step 7: Simplify the equation
Cancelling \( h^2 \) from both sides:
\[ -4h + 4 + k^2 - 8k + 16 = 0 \]
Rearranging gives:
\[ k^2 - 8k - 4h + 20 = 0 \]

### Step 8: Replace variables to get the locus equation
Replacing \( h \) with \( x \) and \( k \) with \( y \):
\[ y^2 - 8y - 4x + 20 = 0 \]

### Final Answer
The locus of the point equidistant from the point (2, 4) and the y-axis is:
\[ y^2 - 4x - 8y + 20 = 0 \]