Question

Write the equation in standard form for the circle that has a diameter with endpoints (6, -9)

Answer 1

Answer:

Step-by-step explanation:

By definition a diameter goes through the center of the circle.  So if we find the midpoint of the diameter, we will have the h and k of the center.  Then we will use one of the points ON the circle as x and y and solve for r.  Here goes:

$M=(\frac{6+(-6)}{2},\frac{-9+(-9)}{2})$ to get a midpoint/center of (0, -9).

The standard form of a circle is

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

Filling in with h, k, x, and y:

$(6-0)^2+(-9-(-9))^2=r^2$ which simplifies to

$6^2+0^2=r^2$  so

$r^2=36$

Now we can write the equation of the circle:

$x^2+(y+9)^2=36$

Answer 2

Answer: x² + (y + 9)² = 36

Step-by-step explanation:

Since we are given the endpoints of the circle, the midpoint between the two endpoints will be the center of the circle. The formula for determining midpoint of a line is

Midpoint = 1/2(x1 + x2), 1/2(y1 + y2)

The circle has a diameter with endpoints (6, -9)

and (-6, -9).

x1 = 6

x2 = - 6

y1 = - 9

y2 = - 9

= 1/2(6 - 6), 1/2(- 9 - 9)

= (0, - 9)

The diameter of the circle is the distance from the endpoint to another endpoint. To determine the diameter of the circle, we would apply The formula for determining the distance between two points on a straight line is expressed as

Distance = √(x2 - x1)² + (y2 - y1)²

Therefore,

Diameter = √(- 6 - 6)² + (- 9 - - 9)²

Diameter = √- 12² + 0² = √144

Diameter = 12

Radius = diameter/2 = 12/2 = 6

The center of the circle is (0, - 8)

The formula for determining the equation of a circle us expressed as

(x - h)² + (y - k)² = r²

Where

r represents the radius of the circle

h and k represents the x and y coordinates of the center of the circle. Comparing with the given points,

h = 0 and k = - 9

Radius, r = 6

Substituting into the formula, it becomes

(x - 0)² + (y - - 9)² = 6²

x² + (y + 9)² = 36