The center of a circle is (h, 7) and the radius is 10. The circle passes through (3, -1). Find all possible values of h.
Scott used the wrong point for the center.
I see it as:
(x-h)^2 + (y-7)^2 = 100
now plug in (3,-1)
(3-h)^2 + (-1-7)^2 = 100
(3-h)^2 = 100 - 64
3-h = ±6
h = 3 ±6
Bonus question: Why did Scott get the same result, even though he had the
incorrect center ?
(h,7) must lie on a circle of radius 10 , centered at (3,-1)
(x - 3)^2 + (y + 1)^2 = 10^2
(h - 3)^2 + (7 + 1)^2 = 10^2 ... (h - 3)^2 = 100 - 64 = 36
h - 3 = ±6
To find the possible values of h, we can use the formula for the distance between two points:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we know that one point on the circle is (h, 7), and the other point is (3, -1). The distance between these two points should be equal to the radius of the circle, which is 10.
Using the formula, we can write the equation as follows:
10 = sqrt((3 - h)^2 + (-1 - 7)^2)
Simplifying the equation:
100 = (3 - h)^2 + (-1 - 7)^2
100 = (3 - h)^2 + (-8)^2
100 = (3 - h)^2 + 64
Rearranging the equation:
(3 - h)^2 = 100 - 64
(3 - h)^2 = 36
Taking the square root of both sides:
3 - h = ±√36
3 - h = ±6
Solving for h:
Case 1:
3 - h = 6
-h = 6 - 3
-h = 3
h = -3
Case 2:
3 - h = -6
-h = -6 - 3
-h = -9
h = 9
Therefore, the two possible values of h are -3 and 9.
To find the possible values of h, we need to determine the equation of the circle using the given information.
The equation of a circle with the center at (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center of the circle is (h, 7) and the radius is 10. Thus, the equation becomes:
(x - h)^2 + (y - 7)^2 = 10^2
The circle passes through the point (3, -1), so we can substitute these coordinates into the equation to find the values of h that satisfy the equation.
(3 - h)^2 + (-1 - 7)^2 = 10^2
(3 - h)^2 + (-8)^2 = 100
(3 - h)^2 + 64 = 100
(3 - h)^2 = 100 - 64
(3 - h)^2 = 36
Taking the square root of both sides, we get:
3 - h = ±6
h = 3 ± 6
Therefore, the possible values of h are:
h = 3 + 6 = 9
h = 3 - 6 = -3
So, the possible values of h are 9 and -3.