What's the equation of the circle? The end points of the diameter (4,8) and (8,-10)

3 Answers
Mar 5, 2018

#(x-6)^2+(y+1)^2=85#

Explanation:

Given diameter end points #(4,8)# and #(8,-10)#
the center of the circle must be at #((4+8)/2,(8-10)/2)=(6,-1)#

The square of the radius can be calculated using the Pythagorean Theorem from this center and the point #(4,8)# as
#color(white)("XXX")r^2=(6-4)^2 +(-1-8)^2=4+81=85#

A congruent circle with center #(0,0)# would have an equationof the form:
#color(white)("XXX")(hatx)^2+(haty)^2=85#

To shift this circle to the required center #(6,-1)#
we need to replace #hatx# with #x-6#
and #haty# with #y+1#

creating the required relation:
#color(white)("XXX")(x-6)^2+(y+1)^2=85#

enter image source here

Mar 5, 2018

Equation of the circle is #(x – 6)^2 + (y +1 )^2 = 85#

Explanation:

The end points of diameter are #(4,8), (8,-10)#

The center-radius form of the circle equation is

#(x – h)^2 + (y – k)^2 = r^2#, with the center being at the point

#(h, k)# and the radius being #r#.

The centre is the mid point of diameter.

So centre is # (4+8)/2,(8-10)/2 or (6,-1)#

#(4,8) and (6,-1) # are the end points of radius #r#.

Distance formula is #D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2#

#:.r^2=(4-6)^2+(8+1)^2 = 85 # . Equation of the circle is

#(x – 6)^2 + (y +1 )^2 = 85#

graph{(x-6)^2+(y+1)^2=85 [-38.9, 38.88, -19.45, 19.44]}

[Ans]

Mar 5, 2018

#(x-6)^2+(y+1)^2=85#

Explanation:

If #(4,-8)# and #(8,-10)# are the end points of the diameter, then the coordinates of the mid point of the diameter are given by:

#((x_1+x_2)/2,(y_1_y_2)/2)#

#((4+8)/2,(8+(-10))/2)#

#(6,-1)#

These are therefore the coordinates of the centre of the circle.

Using the distance formula, we can find the length of the diameter.

The distance formula states that. Distance #bbd# is:

#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#d=sqrt((8-4)^2+((-10)-8)^2)#

#d=sqrt(340)=2sqrt(85)#

If diameter is #2sqrt(85)# then the radius is #(2sqrt(85))/2=sqrt(85)#

General equation of a circle is:

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

Where #k and h# are the #x and y# coordinates of the centre, and #r# is the radius:

Plugging in our values we found previously:

#k=6# , #h=-1# , #r=sqrt(85)#

#(x-6)^2+(y+1)^2=85#

This is the equation of the circle.

Graph:

enter image source here