Circle C has its center at (9,0) and a point is on the circle at A(8,6).
Which answer verifies whether point P(10,−7) lies on the circle?
a. The statement (9−0)2+(8−6)2≠(9−0)2+(10+7)2 is a true statement, so P is not on ⨀C.
b. The equation (9−0)2+(8−6)2=(9−0)2+(10+7)2 is a true statement, so P is on ⨀C.
c. The statement (9−8)2+(0−6)2≠(9−10)2+(0+7)2 is a true statement, so P is not on ⨀C.
d. The equation (9−8)2+(0−6)2=(9−10)2+(0+7)2 is a true statement, so P is on ⨀C.
This question is a good example of how to make a rather simple concept look complicated
centre is ( 9,0) , so equation is
(x-9)^2 + y^2 = r^2
point (8,6) is on it, then 1^2 + 6^2 = r^2 = 37
the circle is : (x-9)^2 + y^2 = 37
Now sub in P(10,−7)
If it satisfies the equation P is on the circle
If it does not satisfy the equation P is not on the circle
I think it is on the line is the answer B?
from where did the 37 came from?
To determine if point P(10, -7) lies on the circle, we need to find the distance between the center of the circle and point P, and then compare it to the radius of the circle.
1. Calculate the distance between the center of the circle and point P using the distance formula:
Distance = √[(x2 - x1)² + (y2 - y1)²]
Plugging in the coordinates of the center (9, 0) and point P (10, -7):
Distance = √[(10 - 9)² + (-7 - 0)²]
= √[1² + (-7)²]
= √[1 + 49]
= √50
= 5√2
2. Compare the calculated distance to the radius of the circle. The radius is the distance between the center of the circle and any point on the circle. In this case, we have a point A(8, 6) on the circle.
Distance between center C and point A:
Distance = √[(8 - 9)² + (6 - 0)²]
= √[(-1)² + 6²]
= √[1 + 36]
= √37
Now, we can compare the two distances:
Radius (distance between center and point on the circle) = √37
Distance between center and point P = 5√2
The point P lies on the circle if the two distances are equal.
Now, let's analyze the answer choices:
a. The statement (9 - 0)² + (8 - 6)² ≠ (9 - 0)² + (10 + 7)² is a true statement, so P is not on circle C.
b. The equation (9 - 0)² + (8 - 6)² = (9 - 0)² + (10 + 7)² is not true, so P is not on circle C.
c. The statement (9 - 8)² + (0 - 6)² ≠ (9 - 10)² + (0 + 7)² is a true statement, so P is not on circle C.
d. The equation (9 - 8)² + (0 - 6)² = (9 - 10)² + (0 + 7)² is not true, so P is not on circle C.
Based on the calculations and analysis, the correct answer is: Option (a) The statement (9 - 0)² + (8 - 6)² ≠ (9 - 0)² + (10 + 7)² is a true statement, so P is not on circle C.