The diameter of a circle has endpoints at (4, 1) and (−6, 1).
What is the x-value of a point on the circle that has a y-value of 4?
The diameter of a circle has endpoints at (4, 1) and (−6, 1). What is the x-value of a point on the circle that has a y-value of 4?
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To find the x-value of a point on the circle given the y-value of 4, we first need to determine the center and radius of the circle.
1. Find the center of the circle. The center is the midpoint of the diameter. The endpoints of the diameter are (4, 1) and (-6, 1):
[
text{Midpoint} = left(frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}right) = left(frac{4 + (-6)}{2}, frac{1 + 1}{2}right) = left(frac{-2}{2}, 1right) = (-1, 1)
]
2. Find the radius of the circle. The radius is half the distance between the endpoints of the diameter. We can use the distance formula:
[
text{Distance} = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} = sqrt{(-6 – 4)^2 + (1 – 1)^2} = sqrt{(-10)^2} = 10
]
Therefore, the radius ( r = frac{10}{2} = 5 ).
3. Write the equation of the circle. With center (-1,