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

Answer:

The answer is below

Step-by-step 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:

$x=\frac{x_1+x_2}{2},y=\frac{y_1+y_2}{2}$.

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

$For\ x\ coordinate:\\2=\frac{-1+5}{2}\\ 2=2.\\For\ y\ coordinate:\\-3y=\frac{y+7}{2}\\ -6y=y+7.\\-6y-y=7\\-7y=y\\y=-1$

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:

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

The radius of the circle is 5 units