Question

On the xy-plane, the center of circle C is at point (3, -2). If the point (10, -2) lies outside of the circle and the point (3, 3) lies inside of the circle, which of the following could be the radius of the circle?

• A. 5
• B. 5.5
• C. 6
• D. 6.5
• E. 7

Hint:

For distance calculations on a coordinate plane, look for shortcuts. If the x-coordinates or y-coordinates are the same, the distance calculation simplifies. For \(d_{\text{in}}\), the x-coordinates were the same, so the distance was just the absolute difference in y-coordinates. For \(d_{\text{out}}\), the y-coordinates were the same, so the distance was the absolute difference in x-coordinates.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question defines a range for the radius of a circle. The radius must be greater than the distance from the center to any point inside the circle, and it must be less than the distance from the center to any point outside the circle.
Step 2: Key Formula or Approach:
We will use the distance formula, \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), to calculate the distance from the center to each of the given points. Let \(r\) be the radius.
Let \(d_{\text{in}}\) be the distance to the inside point. We must have \(r>d_{\text{in}}\).
Let \(d_{\text{out}}\) be the distance to the outside point. We must have \(r<d_{\text{out}}\).
Step 3: Detailed Explanation:
The center of the circle is \(C = (3, -2)\).
The point inside the circle is \(P_{\text{in}} = (3, 3)\). Let's find the distance from the center to this point: \[ d_{\text{in}} = \sqrt{(3 - 3)^2 + (3 - (-2))^2} = \sqrt{0^2 + (3+2)^2} = \sqrt{5^2} = 5 \] Since this point is inside the circle, the radius \(r\) must be greater than this distance: \(r>5\).
The point outside the circle is \(P_{\text{out}} = (10, -2)\). Let's find the distance from the center to this point: \[ d_{\text{out}} = \sqrt{(10 - 3)^2 + (-2 - (-2))^2} = \sqrt{7^2 + 0^2} = \sqrt{7^2} = 7 \] Since this point is outside the circle, the radius \(r\) must be less than this distance: \(r<7\).
Combining these two inequalities, we find the possible range for the radius: \[ 5<r<7 \] Step 4: Final Answer:
We need to find an option that falls between 5 and 7.
(A) 5: Not possible, as \(r\) must be strictly greater than 5.
(B) 5.5: Possible, as \(5<5.5<7\).
(C) 6: Possible, as \(5<6<7\).
(D) 6.5: Possible, as \(5<6.5<7\).
(E) 7: Not possible, as \(r\) must be strictly less than 7.
Since options (B), (C), and (D) are all valid possibilities, and this is a single-choice question, the question may be flawed. However, if we must choose one, any of them could be correct. We select 6.5 as a valid answer.