To find a relation between \( x \) and \( y \) such that the point \( (x, y) \) is equidistant from the points \( (3, 6) \) and \( (-3, 4) \), we can follow these steps:
### Step 1: Use the Distance Formula
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
We will apply this formula to find the distances from the point \( (x, y) \) to the points \( (3, 6) \) and \( (-3, 4) \).
### Step 2: Calculate the Distance from \( (x, y) \) to \( (3, 6) \)
The distance \( d_1 \) from \( (x, y) \) to \( (3, 6) \) is:
\[
d_1 = \sqrt{(3 - x)^2 + (6 - y)^2}
\]
### Step 3: Calculate the Distance from \( (x, y) \) to \( (-3, 4) \)
The distance \( d_2 \) from \( (x, y) \) to \( (-3, 4) \) is:
\[
d_2 = \sqrt{(-3 - x)^2 + (4 - y)^2}
\]
### Step 4: Set the Distances Equal
Since the point \( (x, y) \) is equidistant from both points, we set \( d_1 = d_2 \):
\[
\sqrt{(3 - x)^2 + (6 - y)^2} = \sqrt{(-3 - x)^2 + (4 - y)^2}
\]
### Step 5: Square Both Sides
To eliminate the square roots, we square both sides:
\[
(3 - x)^2 + (6 - y)^2 = (-3 - x)^2 + (4 - y)^2
\]
### Step 6: Expand Both Sides
Expanding both sides using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
- Left Side:
\[
(3 - x)^2 = 9 - 6x + x^2
\]
\[
(6 - y)^2 = 36 - 12y + y^2
\]
Thus, the left side becomes:
\[
9 - 6x + x^2 + 36 - 12y + y^2 = x^2 + y^2 - 6x - 12y + 45
\]
- Right Side:
\[
(-3 - x)^2 = 9 + 6x + x^2
\]
\[
(4 - y)^2 = 16 - 8y + y^2
\]
Thus, the right side becomes:
\[
9 + 6x + x^2 + 16 - 8y + y^2 = x^2 + y^2 + 6x - 8y + 25
\]
### Step 7: Set the Expanded Forms Equal
Now we have:
\[
x^2 + y^2 - 6x - 12y + 45 = x^2 + y^2 + 6x - 8y + 25
\]
### Step 8: Simplify the Equation
Subtract \( x^2 + y^2 \) from both sides:
\[
-6x - 12y + 45 = 6x - 8y + 25
\]
Now, rearranging gives:
\[
-6x - 12y + 8y = 6x + 25 - 45
\]
\[
-6x - 4y = 6x - 20
\]
### Step 9: Combine Like Terms
Combining like terms results in:
\[
-12x - 4y = -20
\]
### Step 10: Divide by -4
Dividing the entire equation by -4 gives:
\[
3x + y = 5
\]
### Final Relation
Thus, the relation between \( x \) and \( y \) is:
\[
3x + y = 5
\]
