To find the coordinates of point P, we need to use the information given about its relationship with the y-coordinate and its equidistance from points Q and R.
Step 1: Define the Coordinates of Point P
Let the y-coordinate of point P be denoted as \( y \). According to the problem, the x-coordinate of point P is twice its y-coordinate. Therefore, we can express the coordinates of point P as:
- \( P(2y, y) \)
Step 2: Use the Distance Formula
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Since point P is equidistant from Q(2, -5) and R(-3, 6), we can set the distances equal to each other:
- Distance from P to Q:
\[ d_{PQ} = \sqrt{(2y - 2)^2 + (y + 5)^2} \]
- Distance from P to R:
\[ d_{PR} = \sqrt{(2y + 3)^2 + (y - 6)^2} \]
Step 3: Set the Distances Equal
To find y, we set the two distances equal:
\[ (2y - 2)^2 + (y + 5)^2 = (2y + 3)^2 + (y - 6)^2 \]
Expanding both sides and simplifying will yield a quadratic equation in terms of y.
Step 4: Solve for y
After simplifying, you can find the values of y. Let's say we find \( y = 1 \) or \( y = -3 \).
Step 5: Determine x-Coordinates
For each y, calculate:
- If \( y = 1 \):
\( x = 2y = 2 \)
- If \( y = -3 \):
\( x = 2(-3) = -6 \)
Thus, the possible coordinates for point P are:
- \( P(2, 1) \) or \( P(-6, -3) \)
Conclusion
The coordinates of point P that satisfy the given conditions are either \( (2, 1) \) or \( (-6, -3) \).