Question

A circle has the equation (x+1)^2+(y-3)^2=16. Find the distance from the center of the circle to the following points.

(2,1)
(4,1)
(3,3)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the distance from the center of the circle to a specific point, we first need to find the coordinates of the center of the circle. The equation of a circle in the form (x-h)^2 + (y-k)^2 = r^2 represents a circle centered at the point (h, k) with radius r.

In the given equation (x+1)^2 + (y-3)^2 = 16, we can see that the center of the circle is at the point (-1, 3) and the radius is 4.

Now we can find the distance from the center of the circle to each of the given points:

1. Distance from center (-1, 3) to point (2, 1):
Distance = √[(2 - (-1))^2 + (1 - 3)^2]
Distance = √[3^2 + (-2)^2]
Distance = √[9 + 4]
Distance = √13

2. Distance from center (-1, 3) to point (4, 1):
Distance = √[(4 - (-1))^2 + (1 - 3)^2]
Distance = √[5^2 + (-2)^2]
Distance = √[25 + 4]
Distance = √29

3. Distance from center (-1, 3) to point (3, 3):
Distance = √[(3 - (-1))^2 + (3 - 3)^2]
Distance = √[4^2 + 0^2]
Distance = √16
Distance = 4

Therefore, the distances from the center of the circle to the given points are:
- √13 (approximately 3.61 units) for point (2, 1)
- √29 (approximately 5.39 units) for point (4, 1)
- 4 units for point (3, 3)

Answer 2

what do these distances tell you about weather these points are inside or outside the circle

Answer 3

[AI-generated answer — Bot GPT 3.5]

The distances we calculated are the distances from the center of the circle to each of the given points.

1. Distance from center (-1, 3) to point (2, 1): Approximately 3.61 units
2. Distance from center (-1, 3) to point (4, 1): Approximately 5.39 units
3. Distance from center (-1, 3) to point (3, 3): 4 units

Since the radius of the circle is 4, we can see that:
- The distance from the center to point (2, 1) is less than the radius (3.61 < 4), so this point is inside the circle.
- The distance from the center to point (4, 1) is greater than the radius (5.39 > 4), so this point is outside the circle.
- The distance from the center to point (3, 3) is equal to the radius (4 = 4), so this point is on the circle itself.

In summary:
- Point (2, 1) is inside the circle.
- Point (4, 1) is outside the circle.
- Point (3, 3) is on the circle.