How do you write an equation for for circle given that the endpoints of the diameter are (-2,7) and (4,-8)?

Question

Theodore Abad · Accepted Answer

#(x-1)^2+(y+1/2)^2=261/4# #"given the endpoints of the diameter then the centre is at"# #"the midpoint and the radius is the distance from the "# #"centre to either of the 2 endpoints"# #"the equation of a circle in standard form is"# #color(red)(bar(ul(|color(white)(2/2)color(black)((x-a)^2+(y-b)^2=r^2)color(white)(2/2)|)))# #"where "(a,b)" are the coordinates of the centre and r "# #"is the radius"# #"midpoint "=[1/2(-2+4),1/2(7-8)]=(1,-1/2)# #"to calculate the radius use the "color(blue)"distance formula"# #•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)# #"let "(x_1,y-1)=(4,-8)" and "(x_2,y_2)=(1,-1/2)# #d=sqrt((1-4)^2+(-1/2+8)^2)# #color(white)(d)=sqrt(9+225/4)=sqrt261/2# #(x-1)^2+(y+1/2)^2=(sqrt261/2)^2# #rArr(x-1)^2+(y+1/2)^2=261/4larrcolor(blue)"equation of circle"#

Aurora Ayala · Answer

Therefore, equation of the given circle is

#color(red)((x - 1)^2 + (y + (1/2))^2 = (8.0777)^2)#

Center coordinates #O ((4-2)/2, (-7+8)/2) = O(1, -(1/2)#

diameter /2 = radius = #r = sqrt((4+2)^2 + (-8-7)^2) /2 = color(blue)(8.0777)#

Standard equation of a circle is

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

Therefore, equation of the given circle is

#color(red)((x - 1)^2 + (y + (1/2))^2 = (8.0777)^2)#

Allison Allen · Answer

undefined To write the equation of a circle given the endpoints of its diameter, you can follow these steps:

1. Find the center of the circle by averaging the coordinates of the endpoints of the diameter.
2. Calculate the radius of the circle by finding the distance between one of the endpoints and the center.
3. Use the center and radius to write the equation of the circle in standard form, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

Let's calculate:

1. Center of the circle:
   $h = \frac{-2 + 4}{2} = 1$
   $k = \frac{7 - 8}{2} = -\frac{1}{2}$

So, the center of the circle is (1, -0.5).

2. Radius of the circle:
   Use the distance formula to find the distance between one of the endpoints and the center:
   $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
   Let's choose (-2, 7) as one of the endpoints:
   $r = \sqrt{(1 - (-2))^2 + ((-0.5) - 7)^2}$
   $r = \sqrt{(3)^2 + (-7.5)^2}$
   $r = \sqrt{9 + 56.25}$
   $r = \sqrt{65.25}$
   $r ≈ 8.08$

3. Equation of the circle:
   $(x - 1)^2 + (y + 0.5)^2 = (8.08)^2$
   $(x - 1)^2 + (y + 0.5)^2 ≈ 65.25$

So, the equation of the circle is $(x - 1)^2 + (y + 0.5)^2 ≈ 65.25$.