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 PP and less than 4 cm from QQ:

  • The points must be inside the circle with radius 3cm3 \, \text{cm} centered at PP and inside the circle with radius 4cm4 \, \text{cm} centered at QQ.
  • 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 PP and less than 4 cm from QQ:

  • The points must lie outside the circle centered at PP with radius 3 \, \text{cm}** and inside the circle centered at \( Q with radius 4cm4 \, \text{cm}.
  • This corresponds to the purple area that is outside PP's circle but inside QQ's circle.

Answer for (b): Diagram D (purple area outside PP but inside QQ).


Would you like a more detailed explanation or breakdown of how the regions were determined?

Follow-up Questions:

  1. How is the locus of points concept used in real-world scenarios like navigation or engineering?
  2. Can you explain why the intersection of two circles represents points satisfying both conditions in Part (a)?
  3. How would the solution change if the radius of one circle were increased or decreased?
  4. What happens to the diagrams if the two centers PP and QQ are moved farther apart?
  5. 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