In the
x
y
−
xy−
plane, the center of a circle has coordinates
(
6
,
10
)
(6,10)
and the circle touches the
y
−
y−
axis at one point only. What is the radius of the circle
To find the radius of the circle, we need to determine the distance between the center of the circle and the point on the y-axis where the circle touches it.
First, let's find the equation of the circle. We know that the center of the circle is located at (6, 10). The general equation of a circle is given by:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the coordinates of the center, and r represents the radius of the circle. In this case, the equation becomes:
(x - 6)^2 + (y - 10)^2 = r^2
Since the circle touches the y-axis at one point only, the x-coordinate of this point is 0. Substituting x = 0 into the equation gives us:
(0 - 6)^2 + (y - 10)^2 = r^2
36 + (y - 10)^2 = r^2
Now, we can solve for y:
(y - 10)^2 = r^2 - 36
y - 10 = ±√(r^2 - 36)
y = 10 ± √(r^2 - 36)
Since the circle touches the y-axis at one point only, the ± sign becomes +:
y = 10 + √(r^2 - 36)
Since this point lies on the y-axis, the x-coordinate is 0, and the y-coordinate is given by the equation above. We can substitute y = 0 into the equation and solve for r:
0 = 10 + √(r^2 - 36)
√(r^2 - 36) = -10
Since r cannot be negative (as it represents the radius), we can drop the negative sign:
√(r^2 - 36) = 10
Square both sides to eliminate the square root:
r^2 - 36 = 100
r^2 = 100 + 36
r^2 = 136
Taking the square root of both sides:
r = √136
r ≈ 11.66
Therefore, the radius of the circle is approximately 11.66 units.
To find the radius of the circle, we need to find the distance between the center of the circle and the point where it touches the y-axis.
Since the point where the circle touches the y-axis is on the y-axis, its x-coordinate is 0. Therefore, the distance between the center of the circle (6, 10) and the point on the y-axis (0, y) is:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
Applying this formula, we have:
distance = √((0 - 6)^2 + (y - 10)^2)
Since the circle touches the y-axis at one point only, the distance is equal to the radius of the circle. Let's call it r. So, we can rewrite the equation as:
r = √((-6)^2 + (y - 10)^2)
Simplifying further, we get:
r = √(36 + (y - 10)^2)
Therefore, the radius of the circle is √(36 + (y - 10)^2).
I will interpret this as:
In the xy-plane, the center of a circle has coordinates
( 6,10) and the circle touches the y-axis at one point only. What is the radius of the circle.
If you make a sketch of the circle that has a centre of (6,10) and touches the y-axis, you can see that the radius has to be 6
It's equation must be (x-6)^2 + (y-10)^2 = 36