1. In the xy-plane, a circle has its center at the point with coordinates (4,3), and the origin lies on the circle. The point with which of the following coordinates also lies on the circle?
A. (-4,-3)
B. (3,4)
C. (6,8)
D. (7,7)
E. (9,8)
How do you do this?
The distance of (4,3) from (0,0) is 5. So, the circle is
(x-4)^2 + (y-3)^2 = = 25
Now, you know that if the center were at (0,0), (±3,±4) and (±4,±3) would lie on the circle.
So, shift those possibilities right and up, and see what drops out.
(7-4)^2 + (7,3)^2 = 3^2+4^2 = 25
Wait, why did you use Pythagoreon theorom for a circle above in your explanation? I don't understand why you do that.
To determine which point lies on the circle, you can calculate the distance between the center of the circle and each of the given points. If the distance is equal to the radius of the circle, then that point lies on the circle.
Here's how you can do it step by step:
1. Find the radius of the circle: The distance between the center of the circle (4,3) and the origin (0,0) can be found using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). Plugging in the values, we get d = sqrt((0 - 4)^2 + (0 - 3)^2) = sqrt((-4)^2 + (-3)^2) = sqrt(16 + 9) = sqrt(25) = 5. So, the radius of the circle is 5.
2. Calculate the distance between the center and each given point: Use the distance formula to find the distance between the center (4,3) and each of the given points.
For option A. (-4,-3):
d = sqrt((-4 - 4)^2 + (-3 - 3)^2) = sqrt((-8)^2 + (-6)^2) = sqrt(64 + 36) = sqrt(100) = 10
For option B. (3,4):
d = sqrt((3 - 4)^2 + (4 - 3)^2) = sqrt((-1)^2 + (1)^2) = sqrt(1 + 1) = sqrt(2)
For option C. (6,8):
d = sqrt((6 - 4)^2 + (8 - 3)^2) = sqrt((2)^2 + (5)^2) = sqrt(4 + 25) = sqrt(29)
For option D. (7,7):
d = sqrt((7 - 4)^2 + (7 - 3)^2) = sqrt((3)^2 + (4)^2) = sqrt(9 + 16) = sqrt(25) = 5
For option E. (9,8):
d = sqrt((9 - 4)^2 + (8 - 3)^2) = sqrt((5)^2 + (5)^2) = sqrt(25 + 25) = sqrt(50)
3. Compare the distances with the radius:
- If the distance is 5 (equal to the radius), then that point lies on the circle.
- If the distance is less than 5, the point is inside the circle.
- If the distance is greater than 5, the point is outside the circle.
Based on the calculations:
- Option A. (-4,-3) has a distance of 10, which is greater than the radius. So, (-4,-3) does not lie on the circle.
- Option B. (3,4) has a distance of sqrt(2), which is less than the radius. So, (3,4) does not lie on the circle.
- Option C. (6,8) has a distance of sqrt(29), which is greater than the radius. So, (6,8) does not lie on the circle.
- Option D. (7,7) has a distance of 5, which is equal to the radius. So, (7,7) lies on the circle.
- Option E. (9,8) has a distance of sqrt(50), which is greater than the radius. So, (9,8) does not lie on the circle.
Therefore, the point with the coordinates (7,7) is the only point that lies on the circle. The answer is D. (7,7).