The point M(4, 3) is given. a) A circle with center M lies completely in one quadrant. What can you say about the radius? b) A circle with center M lies in two quadrants. What can... The point M(4, 3) is given. a) A circle with center M lies completely in one quadrant. What can you say about the radius? b) A circle with center M lies in two quadrants. What can you say about the radius? c) A circle with center M lies in four quadrants. What can you say about the radius?
Understand the Problem
The question is asking for a geometric interpretation involving the position of a circle centered at point M(4, 3) with respect to the quadrants of the coordinate plane. Specifically, it explores conditions under which the circle lies completely in one quadrant, intersects two quadrants, or spans all four quadrants, prompting an analysis of the radius in each case.
Answer
a) $r < 3$ b) $3 \leq r < 4$ c) $r \geq 5$
Answer for screen readers
a) The radius must be less than 3.
b) The radius must be at least 3 and less than 4.
c) The radius must be at least 5.
Steps to Solve
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Circle Completely in One Quadrant To have a circle with center $M(4, 3)$ lie completely in one quadrant, the radius must be less than the distance from the center to the edges of that quadrant. The x-coordinate is 4 (right of y-axis), and the y-coordinate is 3 (above x-axis). This means:
- For the first quadrant, the radius must be less than $\min(4, 3) = 3$.
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Circle in Two Quadrants For a circle centered at $M(4, 3)$ to lie in two quadrants, the radius must be greater than or equal to the distance to reach the x-axis and simultaneously less than the distance to the first quadrant's edge.
- The radius can be equal to 3 (to touch the x-axis) but less than 4 (to stay away from the y-axis), which means: $3 \leq r < 4$.
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Circle Spanning All Four Quadrants For the circle centered at $M(4, 3)$ to span all four quadrants, the radius must be at least the distance to both axes.
- The required radius is $r \geq 5$, since it must exceed both the distance to the x-axis (3) and the y-axis (4): $$ r \geq \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 $$
a) The radius must be less than 3.
b) The radius must be at least 3 and less than 4.
c) The radius must be at least 5.
More Information
The geometric interpretation of the circle's radius relative to its center position helps visualize how a circle interacts with the coordinate plane's quadrants. Understanding these relationships can be useful in various applications, from computer graphics to physics.
Tips
- Confusing the distances: It's important to clearly differentiate the distances to the x-axis and y-axis when assessing the radius limits.
- Forgetting to include both directions: When discussing placement in two quadrants, ensure both dimensions (x and y) are considered.
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