A circle has its centre on the x-axis and passes through(4,1)and(1,5).find its equation.

You know that

(x-h)^2 + (y-k)^2 = r^2 is the equation of a general circle.

Since the center is on the x-axis, k=0, and you have

(x-h)^2 + y^2 = r^2

Plug in your numbers to get

(4-h)^2 + 1 = r^2
(1-h)^2 + 25 = r^2

That gives you
h = -3/2
r^2 = 125/4

so,

(x + 3/2)^2 + y^2 = 125/4

Why did the circle cross the x-axis? Because it wanted to center itself!

Now, let's find the equation for this clown circle.
The center of the circle lies on the x-axis, so its y-coordinate is 0.

Using the midpoint formula, we can find the x-coordinate of the center:
(x1 + x2)/2 = (4 + 1)/2 = 5/2

Therefore, the center of the circle is (5/2, 0).

To find the radius, we can use the distance formula between the center and any point on the circle:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let's choose (4, 1) as the other point for our calculations:
r = sqrt((4 - 5/2)^2 + (1 - 0)^2)
= sqrt((-1/2)^2 + 1^2)
= sqrt(1/4 + 1)
= sqrt(5/4)
= sqrt(5)/2

Thus, the radius is sqrt(5)/2.

Putting all this information together, the equation of the circle becomes:
(x - 5/2)^2 + y^2 = (sqrt(5)/2)^2

Simplifying that, we get:
(x - 5/2)^2 + y^2 = 5/4

So, the equation of the circle is x^2 - 5x + 25/4 + y^2 = 5/4.

Hope this clowning explanation brightened your day!

To find the equation of a circle, we need to determine the center and radius of the circle from the given information.

Step 1: Determine the center of the circle.
Since the circle has its center on the x-axis, the y-coordinate of the center is 0.

Step 2: Find the midpoint of the two given points.
The midpoint of two points is calculated by taking the average of their x-coordinates and their y-coordinates.

For (4, 1) and (1, 5):

Midpoint of X-coordinate = (4 + 1)/2 = 2.5
Midpoint of Y-coordinate = (1 + 5)/2 = 3

So, the coordinates of the midpoint are (2.5, 3).

Step 3: Determine the radius of the circle.
The radius of the circle can be found by calculating the distance between the center of the circle and any one of the given points. We can use the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

For example, if we choose (4, 1):
Distance = sqrt((4 - 2.5)^2 + (1 - 0)^2)
= sqrt(1.5^2 + 1^2)
= sqrt(2.25 + 1)
= sqrt(3.25)
= 1.802

So, the radius of the circle is approximately 1.802.

Step 4: Write the equation of the circle.
The equation of a circle with its center at (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2.

Using the midpoint (2.5, 3) as the center and the radius 1.802, the equation of the circle is:

(x - 2.5)^2 + (y - 0)^2 = (1.802)^2

Simplifying further:

(x - 2.5)^2 + y^2 = 3.247204

So, the equation of the circle passing through (4, 1) and (1, 5) is (x - 2.5)^2 + y^2 = 3.247204.

To find the equation of a circle, we need two important pieces of information: the center of the circle and its radius.

Let's start by finding the center of the circle. Since the center lies on the x-axis, its y-coordinate will be zero. Let's denote the x-coordinate of the center as "a." Therefore, the center of the circle is (a, 0).

Now, let's find the distance between the center of the circle and one of the given points on the circumference of the circle, (4, 1). We can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the values, we have:

Distance = √((4 - a)^2 + (1 - 0)^2)

To find the radius, we need to calculate the distance between the center (a, 0) and any point on the circumference of the circle. The given point (1, 5) can be used:

Radius = √((1 - a)^2 + (5 - 0)^2)

Since the radius is the same as the distance, we have:

√((4 - a)^2 + (1 - 0)^2) = √((1 - a)^2 + (5 - 0)^2)

Squaring both sides of the equation, we get:

(4 - a)^2 + 1 = (1 - a)^2 + 25

Expanding the squares, we have:

(16 - 8a + a^2) + 1 = (1 - 2a + a^2) + 25

Combining like terms, we get:

17 - 8a + a^2 = 26 - 2a + a^2

Moving all the terms to one side, we have:

17 - 26 = -2a + 8a

-9 = 6a

Dividing both sides by 6, we get:

a = -9/6 = -3/2

Therefore, the x-coordinate of the center of the circle is -3/2, and the y-coordinate is 0. So the center of the circle is (-3/2, 0).

Now, to find the equation of the circle, we use the standard form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

Plugging in the values, we have:

(x - (-3/2))^2 + (y - 0)^2 = ((1 - (-3/2))^2 + (5 - 0)^2)

Simplifying further, we have:

(x + 3/2)^2 + y^2 = (5/2)^2 + 25

(x + 3/2)^2 + y^2 = 25/4 + 100/4

(x + 3/2)^2 + y^2 = 125/4

Therefore, the equation of the circle is (x + 3/2)^2 + y^2 = 125/4.