Let's analyze the problem step by step:
1. Original Circle:
- Radius: [tex]\( 3 \)[/tex]
- Center: [tex]\( (0, 0) \)[/tex]
2. Modified Circle:
- Radius: [tex]\( 3 \times 3 = 9 \)[/tex]
- Center: [tex]\( (0, 0) \)[/tex]
We need to determine which of the given points will NOT be inside this new circle.
3. Calculate the Distance:
To determine if a point [tex]\((x, y)\)[/tex] lies within the circle of radius 9 centered at the origin, we will calculate the Euclidean distance from the origin to the point:
[tex]\[
d = \sqrt{x^2 + y^2}
\][/tex]
4. Check Each Point:
- Point [tex]\((-9, 3)\)[/tex]:
[tex]\[
d = \sqrt{(-9)^2 + 3^2} = \sqrt{81 + 9} = \sqrt{90} \approx 9.4868
\][/tex]
Since [tex]\( 9.4868 > 9 \)[/tex], the point [tex]\((-9, 3)\)[/tex] is NOT inside the circle.
- Point [tex]\((-4, 3)\)[/tex]:
[tex]\[
d = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\][/tex]
Since [tex]\( 5 < 9 \)[/tex], the point [tex]\((-4, 3)\)[/tex] is inside the circle.
- Point [tex]\((2, -4)\)[/tex]:
[tex]\[
d = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.4721
\][/tex]
Since [tex]\( 4.4721 < 9 \)[/tex], the point [tex]\((2, -4)\)[/tex] is inside the circle.
- Point [tex]\((3, 7)\)[/tex]:
[tex]\[
d = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.6158
\][/tex]
Since [tex]\( 7.6158 < 9 \)[/tex], the point [tex]\((3, 7)\)[/tex] is inside the circle.
5. Conclusion:
The point [tex]\((-9, 3)\)[/tex] is NOT inside the circle with radius 9 centered at the origin. The other points, [tex]\((-4, 3)\)[/tex], [tex]\((2, -4)\)[/tex], and [tex]\((3, 7)\)[/tex] are inside the circle.
