Question

How do you write the equation of the circle with a diameter that has endpoints at (7, –4) and (1, –10)?

Answer 1

`(x - 4)^2 + (y + 7)^2 = 18`

Explanation:

The standard form of the equation of a circle is.

`color(red)(|bar(ul(color(white)(a/a)color(black)( (x - a)^2 + (y - b)^2 = r^2)color(white)(a/a)|)))`

where (a ,b ) are the coordinates of the centre and r , the radius.

To obtain the equation , we require to find it's centre and radius.

Given the endpoints of the diameter , the centre will be at the mid-point and the radius will be the distance from the centre to either one of the endpoints.

Given 2 points `A (x_1,y_1)" and "B (x_2,y_2) " the mid-point is "`

`color(blue)(|bar(ul(color(white)(a/a)color(black)( M_(AB) = 1/2(x_1 + x_2) , 1/2(y_1 + y_2))|)))`

Here the 2 endpoints are (7,-4) and (1,-10)

hence centre = `[ 1/2(7+1) , 1/2(-4-10) ] = (4 , -7)`

To calculate the radius: distance from (4,-7) to (7,-4) using the `color(blue)" distance formula "`

`color(green)(|bar(ul(color(white)(a/a)color(black)( d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points "`

let `(x_1,y_1)=(7,-4)" and " (x_2,y_2) =(4,-7)`

`rArr " radius " = sqrt((4-7)^2 + (-7+4)^2) = sqrt(9+9)`
`= sqrt18`

We now have centre = (4 , -7) and radius =`sqrt18`
hence equation of circle:

`(x - 4)^2 + (y + 7)^2 = 18`