How do you write the equation of the circle with endpoints of its diameter at (-4,7) and (8,-9)?
2 Answers
Explanation:
#"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"#
Explanation:
If endpoint of the diameter are
then the circle has
- a center at
#(color(red)(x_c),color(blue)(y_c))=((-4+8)/2,(7+(-9))/2)=(color(red)2,color(blue)(-1))#
and - a diameter of
#color(green)d=sqrt((-4-(+8))^2+(7-(-9))^2)=sqrt(12^2+16^2)=sqrt(144+256)=sqrt(400)=20#
#rarr# a radius of#color(magenta)r=color(green)d/2=color(magenta)10#
The general equation of a circle with center
In this case the equation will be: