How do you write the equation of the circle with endpoints of its diameter at (-4,7) and (8,-9)?

Question

Ella Armstrong · Accepted Answer

#(x-2)^2+(y+1)^2=100# #"the standard form of the equation of a circle 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"# #"to find the centre we require the "color(blue)"midpoint "" of the"# #"2 given points"# #"centre "=[1/2(-4+8),1/2(7-9)]# #color(white)(centre)=(2,-1)# #"the radius is the distance from the centre to either of "# #"the 2 given points"# #"calculate the radius using 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)=(2,-1)" and "(x_2,y_2)=(-4,7)# #r=sqrt((-4-2)^2+(7+1)^2)=sqrt(36+64)=10# #rArr(x-2)^2+(y-(-1))^2=10^2# #rArr(x-2)^2+(y+1)^2=100larr" equation of circle"#

Autumn Armstrong · Answer

#(x-2)^2+(x+1)^2=100# If endpoint of the diameter are #(-4,7)# and #(8,-9)# then the circle has The general equation of a circle with center #(color(red)(x_c),color(blue)(y_c))# and radius #color(magenta)r# is #color(white)("XXX")(x-color(red)(x_c))^2+(y-color(blue)(y_c))^2=color(magenta)r^2# In this case the equation will be: #color(white)("XXX")(x-2)^2+(x+1)^2=100#

Mason Adams · Answer

undefined The equation of the circle with endpoints of its diameter at (-4,7) and (8,-9) can be written as:

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

where $(h, k)$ is the center of the circle and $r$ is the radius. To find the center, use the midpoint formula:

$$h = \frac{x_1 + x_2}{2}$$
$$k = \frac{y_1 + y_2}{2}$$

Plugging in the values:

$$h = \frac{-4 + 8}{2} = 2$$
$$k = \frac{7 - 9}{2} = -1$$

Next, calculate the radius using the distance formula:

$$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Plugging in the values:

$$r = \sqrt{(8 - (-4))^2 + (-9 - 7)^2}$$
$$r = \sqrt{12^2 + (-16)^2}$$
$$r = \sqrt{144 + 256}$$
$$r = \sqrt{400}$$
$$r = 20$$

Therefore, the equation of the circle is:

$$(x - 2)^2 + (y + 1)^2 = 20^2$$