# How do we find the standard equation of the circle with endpoints of a diameter #(1/2, 4)# #(3/2, -1)#?

##### 2 Answers

#### Explanation:

The standard equation of a circle takes the form:

We need to find the radius,

The length of the diameter is given by:

The radius will be half the diameter,

We can then use the midpoint formula to find the coordinates of the centre. The midpoint formula simply averages the values of the endpoints:

Now we have all the information we need to write the equation for the circle:

#### 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 the radius.

#"the centre will be at the midpoint of the diameter"#

#"midpoint "=[1/2(1/2+3/2),1/2(4-1)]"#

#rArr"centre " =(1,3/2)larr(a,b)#

#"the radius is the distance from the centre to either of "#

#"the endpoints"#

#"to calculate r use the "color(blue)"distance formula"#

#"points are " (1,3/2)" and " (3/2,-1)#

#r=sqrt((3/2-1)^2+(-1-3/2)^2)#

#color(white)(r)=sqrt(1/4+25/4)=sqrt(13/2)#

#rArr(x-1)^2+(y-3/2)^2=(sqrt(13/2))^2#

#rArr(x-1)^2+(y-3/2)^2=13/2" is the equation"#