Math Problem Statement
Find a point on the y-axis that is equidistant from the points (8, −8) and (3, 3).
Solution
To find a point on the y-axis that is equidistant from the points and , we need to determine the coordinates of the point on the y-axis, which will have the form .
Step 1: Use the distance formula
The distance between two points and is given by the distance formula:
Step 2: Set up two distance equations
Let the point on the y-axis be .
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The distance between and is:
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The distance between and is:
Since the point on the y-axis is equidistant from both and , we set .
Step 3: Set up the equation
Step 4: Square both sides
Squaring both sides of the equation to eliminate the square roots:
Step 5: Expand both sides
Expand the squares on both sides:
Simplifying:
Step 6: Eliminate and simplify
Since appears on both sides, we can subtract it from both sides:
Now, combine like terms:
Step 7: Solve for
Subtract 128 from both sides:
Now, divide by 22:
Step 8: Conclusion
The point on the y-axis that is equidistant from and is .
Would you like further details on this solution? Here are five related questions to explore:
- How can you use the midpoint formula to check if the point lies on the perpendicular bisector of the two given points?
- What is the significance of symmetry when finding points equidistant from two given points?
- How would the problem change if the point was restricted to the x-axis instead of the y-axis?
- Can you find the equation of the circle that passes through the points and with center ?
- How does this method generalize for finding points equidistant from two points in three-dimensional space?
Tip: When solving distance problems, always use the symmetry of the coordinate axes to simplify the computation process.
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Math Problem Analysis
Mathematical Concepts
Geometry
Distance Formula
Coordinate Geometry
Algebra
Formulas
Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Theorems
Equidistance theorem: A point is equidistant from two other points if its distances to both points are equal.
Suitable Grade Level
Grades 9-11
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