There exist two circles that go through two points (1,3) (2,4) and are tangent to the y-axis. Letting the radii of the circles be A and B, implies that A*B = ?

Question

There exist two circles that go through two points (1,3) (2,4) and are tangent to the y-axis. Letting the radii of the circles be  A and B, implies that A*B = ?

diamonds4242 · Answer

The perpendicular bisector of the line segment between the given points will contain the centers of the two circles. That bisector line is 
.. y = 5 - x 

The distance from a point on the line (x, 5-x) to one of the points will equal the distance from the point on the line to the y-axis. 
.. (x - 1)^2 + ((5-x) - 3)^2 = x^2 
.. x^2 - 2x + 1 + 25 - 10x + x^2 - 6(5 - x) + 9 = x^2 … expand the left side 
.. x^2 - 6x + 5 = 0 … express in standard form 
.. (x-5)(x-1) = 0 … factor 
.. x = {1, 5} = {a, b} … these are the x-coordinates of the circle centers, and also their radii 

a*b = 1*5 =5

diamonds4242 · Answer

I hope that helped :)

styxer · Answer

@Diamonds4242 I see, thank you very much

diamonds4242 · Answer

Awe your welcome! :)