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