Math Problem Statement
What is the locus of points less than 3 cm from P and less than 4 cm from Q? What is the locus of points more than 3 cm from P and less than 4 cm from Q?
Solution
To answer the problem regarding the locus of points:
Part (a): Less than 3 cm from and less than 4 cm from :
- The points must be inside the circle with radius centered at and inside the circle with radius centered at .
- This means the solution is the overlap (intersection) of the two circles.
Answer for (a): Diagram F (purple overlap region inside both circles).
Part (b): More than 3 cm from and less than 4 cm from :
- The points must lie outside the circle centered at with radius 3 \, \text{cm}** and inside the circle centered at \( Q with radius .
- This corresponds to the purple area that is outside 's circle but inside 's circle.
Answer for (b): Diagram D (purple area outside but inside ).
Would you like a more detailed explanation or breakdown of how the regions were determined?
Follow-up Questions:
- How is the locus of points concept used in real-world scenarios like navigation or engineering?
- Can you explain why the intersection of two circles represents points satisfying both conditions in Part (a)?
- How would the solution change if the radius of one circle were increased or decreased?
- What happens to the diagrams if the two centers and are moved farther apart?
- Can you generalize this problem to three-dimensional spaces with spheres?
Tip: For solving locus problems, always visualize the conditions as regions on a diagram to identify intersections or exclusions!
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Math Problem Analysis
Mathematical Concepts
Geometry
Locus of Points
Circles
Intersection of Sets
Formulas
Equation of a circle: (x - h)^2 + (y - k)^2 = r^2
Theorems
Intersection of two circles
Geometrical properties of circles
Suitable Grade Level
Grades 9-12