Find an equation of the circle
Center on line y–4=0, tangent x-axis at (–2, 0)
the radius is 4
the center lies at (-2,4)
(x+2)^2 + (y-4)^2 = 16
Well, well, well, let me put on my mathematical clown nose and see what we can come up with!
The equation of the circle with its center on the line y - 4 = 0 can be written as (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents its radius.
We know that the center is on the line y - 4 = 0, which means the y-coordinate of the center is 4. So now we have (x - h)^2 + (y - 4)^2 = r^2.
The circle is tangent to the x-axis at (-2, 0). That means the distance from the center (h, 4) to the point (-2, 0) is equal to the radius (r). Using the distance formula, we get:
√[(h - (-2))^2 + (4 - 0)^2] = r
Simplifying this equation, we have √[(h + 2)^2 + 16] = r.
So, the equation of the circle is (x - h)^2 + (y - 4)^2 = (√[(h + 2)^2 + 16])^2.
Now, go ahead and substitute any value you want for h in that equation, and you've found yourself an equation of the circle center on the line y - 4 = 0 and tangent to the x-axis at (-2, 0). Just be careful not to step on any clown shoes while doing the math!
To find the equation of a circle with a center on a given line and tangent to the x-axis, we can follow these steps:
Step 1: Determine the coordinates of the circle's center.
Given that the center is on the line y - 4 = 0, we set y - 4 = 0 and solve for y:
y - 4 = 0
y = 4
So, the y-coordinate of the center is 4.
Step 2: Determine the radius of the circle.
Since the circle is tangent to the x-axis at (-2, 0), the distance from the center (4, y) to (-2, 0) is equal to the radius. So we calculate the distance between these two points using the distance formula:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
r = sqrt[(-2 - 4)^2 + (0 - 4)^2]
r = sqrt[(-6)^2 + (-4)^2]
r = sqrt[36 + 16]
r = sqrt(52)
r = 2sqrt(13)
Step 3: Write the equation of the circle.
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Substituting the values h = 4, k = 4, and r = 2sqrt(13) from steps 1 and 2, we have:
(x - 4)^2 + (y - 4)^2 = (2sqrt(13))^2
(x - 4)^2 + (y - 4)^2 = 4 * 13
(x - 4)^2 + (y - 4)^2 = 52
Therefore, the equation of the circle is (x - 4)^2 + (y - 4)^2 = 52.
To find the equation of a circle, we need the coordinates of its center and the radius. In this case, the center of the circle lies on the line y - 4 = 0 and is tangent to the x-axis at the point (-2, 0).
First, let's determine the center of the circle. Since the center lies on the line y - 4 = 0, we can set y - 4 equal to 0 and solve for x.
y - 4 = 0
y = 4
So, the center of the circle is (x, 4).
Next, let's determine the radius of the circle. We know that the circle is tangent to the x-axis at the point (-2, 0). The distance from the center of the circle to the point of tangency gives us the radius.
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the coordinates of the center (x, 4) and the point of tangency (-2, 0), we have:
r = √((-2 - x)^2 + (0 - 4)^2)
r = √((-2 - x)^2 + 16)
r = √(4 + 4x + x^2 + 16)
r = √(x^2 + 4x + 20)
Now, we have the center of the circle as (x, 4) and the radius as √(x^2 + 4x + 20). Therefore, the equation of the circle is:
(x - x-coordinate of center)^2 + (y - y-coordinate of center)^2 = radius^2
Substituting the values, we get:
(x - x)^2 + (y - 4)^2 = (x^2 + 4x + 20)
Simplifying the equation, we have:
(y - 4)^2 = x^2 + 4x + 20
Therefore, the equation of the circle is (y - 4)^2 = x^2 + 4x + 20.