Write the standard form of the equation of the circle with the given characteristics. endpoints of a diameter: (- 1, 4) , (7, 6)

Answer 1

First, since you know the diameter endpoints, you can determine the center of the circle, which is the midpoint between those two points.

midpoint formula ((x₁+x₂)/2, (y₁+y₂₎/2)

Center of the circle is at (3,5)

So the equation of the circle will have the form (x-3)²+(y-5)²=R² where R is the radius of the circle.

In order to find the radius, you need to find the distance between the center of the circle and a point on the circle. So let's find the distance between (3,5) and (7,6)

Using the distance formula, you find the radius R as R=√((7-3)²+(6-5)²) you should have enough to finish the problem

Answer 2

If (-1, 4) and (7, 6) are the endpoints of the diameter, the midpoints of those would have to be the radius. Using the midpoint formula (you average the x values and y values to find the midpoint), you get the center to be (3, 5)

Using the equations (x - h)^2 + (y - k)^2 = r^2, we can plug in (3,5) for (h,k).

(x - 3)^2 + (y - 5)^2 = r^2. Since we know two points on the circle (the diameter endpoints), we can plug in one of those for x and y to solve for r.

(7 - 3)^2 + (6 - 5)^2 = r^2

4^2 + 1^2 = r^2

17 = r^2

Our final answer is:

(x - 3)^2 + (y - 6)^2 = 17

I hope this helps! Please let me know if you have any questions.