Points `A(-1, y)` and `B(5,7)` lie on a circle with centre `O(2,-3y).` Find the values of `y.` Hence, find the radius of the circle.

Options

A. 2 units
B. 3 units
C. 4 units
D. 5 units

Answer

D

Explanation

Since the radii of a circle are equal, we have
`OA = OB rArr OA^(2) = OB^(2)`
`rArr (2+1)^(2) + (-3y-y)^(2) = (2-5)^(2) + (-3y-7)^(2)`
`rArr 3^(2) + (-4y)^(2) = (-3)^(2) + (3y +7)^(2)`
`rArr 9+16y^(2) = 9+9Y^(2) + 49 +42y`
`rArr 7y^(2)-42y -49 =0`
`y^(2)-6y-7 =0`
`rArr y^(2) -7y +y-7 =0`
`rArr y(y-7) + (y-7) = 0 rArr (y-7)(y+1) = 0`
`rArr y-7 = 0 "or" y +1 = 0 rArr y = 7 "or" y = -1.`
`therefore ` centre is either (2, 3) or (2, -21).
Case I When centre is (2, 3)
In this case, radius `=sqrt((5-2)^(2) + (7-3)^(2)) = sqrt(3^(2) + 4^(2))`
`= sqrt(9 +16) = sqrt(25) = 5` units.
Case II When centre is (2, -21)
In this case, radius ` = sqrt((5-2)^(2) + {7-(-21)}^(2))`
` = sqrt(3^(2) + (28)^(2)) = sqrt(9+784)`
` =sqrt(793) = 28.1` units.