A scientist is plotting the circular path of a partial on a coordinate plane for a lab experiment. The scientist knows the path is a perfect circle and that the particles start at ordered pair (-5, -11). When a particle is halfway down the circle, the particle is the ordered pair (11, 19). The segment formed by connecting these points has a center of the circle midpoint. a) What is the ordered pair that represents the center of the circle? b) What is the length of the radius, in units, of the circle? c) Explain why the the particle can never pass through a point with an x-coordinate of 24 as long as it stays on a circular path.
dang! forgot to divide 6 by 2.
Go with mathhelper's usual careful work.
" start at ordered pair (-5, -11). When a particle is halfway down the circle, the particle is the ordered pair (11, 19). "
This tells me that the two points in the above sentence form the diameter
of the circle, so the center would be midpoint, which is
( (-5+11)/2 , (-11+19)/2 )
= (3,4)
so the circle is (x-3)^2 + (y-4)^2 = r^2
but (-5,-11) lies on this, so
(-5-3)^2 + (-11-4)^2 = r^2
289 = r^2
r = 17
circle is (x-3)^2 + (y-4)^2 = 289
if x = 24
441 + (y-4)^2 = 289
(y-4)^2 = -152
so y-4 is not a real number, thus y is not a real number
To find the center of the circle (a) and the length of the radius (b), we can use the midpoint formula and the distance formula.
a) The center of a circle is the midpoint of any diameter. In this case, we can use the midpoint formula to find the center.
The midpoint formula is:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Given that the particle starts at (-5, -11) and is halfway down the circle at (11, 19), we can plug in these values into the midpoint formula:
Midpoint = ((-5 + 11) / 2, (-11 + 19) / 2)
= (6 / 2, 8 / 2)
= (3, 4)
Therefore, the ordered pair that represents the center of the circle is (3, 4).
b) The distance between the center of the circle and any point on the circumference is the radius of the circle. In this case, we can use the distance formula to find the length of the radius.
The distance formula is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Given that the center of the circle is (3, 4) and a point on the circumference is (-5, -11), we can plug in these values into the distance formula:
Distance = sqrt((-5 - 3)^2 + (-11 - 4)^2)
= sqrt((-8)^2 + (-15)^2)
= sqrt(64 + 225)
= sqrt(289)
= 17
Therefore, the length of the radius, in units, of the circle is 17.
c) The particle can never pass through a point with an x-coordinate of 24 as long as it stays on a circular path. This is because the x-coordinate of the center of the circle is 3, and the radius is 17. Since the x-coordinate of the center is less than 24 and the radius is a fixed length of 17, the point with an x-coordinate of 24 would lie outside the circle. The particle can only travel along the circumference of the circle, so it cannot pass through a point outside the circle.
(a) the midpoint is just the average of the two end points: (6,4)
(b) the radius is half the length of the line segment. Use your normal distance formula.
(c) you have the center and radius. You can see that x=24 is too far from the center.