Write the equation of a circle given the following:

Diameter measures 15 units and the center is at the intersection of y =x + 7 and y= 2x-5

To find the equation of a circle, we need to know the center coordinates and the radius or diameter of the circle.

In this case, we are given that the center of the circle is at the intersection of the lines y = x + 7 and y = 2x - 5.

To find the center coordinates, we can equate the equations of the two lines and solve for x and y:

x + 7 = 2x - 5

Bringing like terms together, we get:

x - 2x = -5 - 7

Simplifying this equation, we have:

-x = -12

Multiplying both sides by -1, we get:

x = 12

Substituting the value of x into either of the original equations, we can solve for y:

y = 12 + 7
y = 19

So, the coordinates of the center are (12, 19).

Next, we are given that the diameter measures 15 units. In a circle, the diameter is twice the length of the radius. So, the radius of this circle is 15/2 = 7.5 units.

Finally, we can write the equation of the circle using the center coordinates and the radius. The equation of a circle with center (h, k) and radius r is given by:

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

Plugging in the values we found, we have:

(x - 12)^2 + (y - 19)^2 = 7.5^2

Therefore, the equation of the circle is:

(x - 12)^2 + (y - 19)^2 = 56.25

To find the equation of a circle given the diameter and the center, we can use the standard form equation of a circle:

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

where (h, k) represents the center of the circle and r represents the radius.

Let's begin by finding the center of the circle. Since the center is at the intersection of two lines, we need to solve the system of equations y = x + 7 and y = 2x - 5.

Setting the equations equal to each other, we have:

x + 7 = 2x - 5

Simplifying, we get:

x - 2x = -5 - 7
-x = -12
x = 12

Substituting x = 12 into either equation, let's use y = x + 7:

y = 12 + 7
y = 19

So the center of the circle is (12, 19).

Next, let's find the radius. Given that the diameter measures 15 units, we can divide it by 2 to get the radius:

radius = diameter / 2 = 15 / 2 = 7.5

Now we have the center (h, k) = (12, 19) and the radius r = 7.5. Plugging these values into the standard form equation of a circle, we get:

(x - 12)^2 + (y - 19)^2 = (7.5)^2

Simplifying, the equation of the circle is:

(x - 12)^2 + (y - 19)^2 = 56.25