Alternative way to find centre (should you forget about the properties of the perpendicular bisector of chord) :

Let coordinates of centre be (a,b)

Now radius is always the same from any point on the circle's circumference to the centre.

So length from each point to the centre is the same.


Use formula Length = √[(x2 - x1)² + (y2 - y1)²]

√[(0 - a)²+(5 + 2√21 - b)²] = √[(0 - a)²+(5 - 2√21 - b)²]

We can square both sides. Expression becomes

a² + (5 + 2√21 - b)² = a² + (5 - 2√21 - b)²


(5 + 2√21)² - 2b(5 + 2√21) + b² = (5 - 2√21)² - 2b(5 - 2√21) + b²


5² + 20√21 + 84 - 10b - 4b√21 = 5² - 20√21 + 84 - 10b + 4b√21


8b√21 = 40√21

b = 5


So the y coordinate is 5.