Question
A circle is centered at \( J(3,3) \) and has a radius of 12 . Where does the point \( F(-6,-5) \) lie? Choose 1 answer: (A) Inside the circle (B) On the circle (C) Outside the circle
Ask by Burton Ray. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The point \( F(-6, -5) \) lies outside the circle.
Solution
To determine where the point \( F(-6, -5) \) lies relative to the circle centered at \( J(3, 3) \) with a radius of 12, follow these steps:
1. **Calculate the Distance Between \( F \) and \( J \):**
Use the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the given points:
\[
\text{Distance} = \sqrt{(-6 - 3)^2 + (-5 - 3)^2} = \sqrt{(-9)^2 + (-8)^2} = \sqrt{81 + 64} = \sqrt{145} \approx 12.04
\]
2. **Compare the Distance to the Radius:**
- If the distance is **less than** 12, \( F \) is **inside** the circle.
- If the distance is **equal to** 12, \( F \) is **on** the circle.
- If the distance is **greater than** 12, \( F \) is **outside** the circle.
Here, \( 12.04 > 12 \), so the point \( F \) lies **outside** the circle.
**Answer:**
**(C) Outside the circle**
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find out where the point \( F(-6,-5) \) lies in relation to the circle centered at \( J(3,3) \) with a radius of 12, we first calculate the distance between points \( J \) and \( F \). The distance formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates: \[ d = \sqrt{(-6 - 3)^2 + (-5 - 3)^2} = \sqrt{(-9)^2 + (-8)^2} = \sqrt{81 + 64} = \sqrt{145} \] Now, since the radius of the circle is 12, we compare \( \sqrt{145} \) to 12. Calculating \( 12^2 \) gives 144. Since \( 145 > 144 \), we find that \( d > 12 \). Therefore, the point \( F(-6,-5) \) lies outside the circle. The answer is (C) Outside the circle.
