Question

Points A(-l, y) and B(5,7) lie on a circle with centre 0(2, -3y). Find the values of y. Hence, find the radius of the circle

Answer 1

The answer is below

Explanation:

Points A(-l, y) and B(5,7) lie on a circle with centre O(2, -3y). This means that AB is the diameter of the circle and OA = OB = radius.

For two points X(
`x_1,y_1`) and Y(
`x_2, y_2`), the coordinates of the midpoint (x, y) between the two points is given as:

`x=(x_1+x_2)/(2),y=(y_1+y_2)/(2)`.

For A(-l, y) and B(5,7) with center O(2, -3y), the value of y can be gotten by:

`For\ x\ coordinate:\\2=(-1+5)/(2)\\ 2=2.\\For\ y\ coordinate:\\-3y=(y+7)/(2)\\ -6y=y+7.\\-6y-y=7\\-7y=y\\y=-1`

The value of y is -1. Therefore A is at (-1, -1) and O is at (2, -3(-1))= (2, 3)

The radius of the circle = OA. The distance between two points X(
`x_1,y_1`) and Y(
`x_2, y_2`) is given as:

`|OX|=√((x_2-x_1)^2+(y_2-y_1)^2) \\\\Therefore\ the\ radius \ |OA|\ is :\\|OA|=√((2-(-1))^2+(3-(-1))^2)=√(25)=5`

The radius of the circle is 5 units