# Circle touching the $y$-axis passing through two points

How to find the equation of the circle touching the $y$-axis given that it passes through two particular points?

• What solution do you have in mind when the two points are $(1,0)$ and $(-1,0)$? – Eric Towers Jan 26 '14 at 8:49
• It is customary to ask you to show some effort, ask what have you done, etc. I gave a some hints below anyway … but please do read this. – Harald Hanche-Olsen Jan 26 '14 at 8:50

Hint: The center of the circle will be equidistant from the $y$ axis and either of the two centers. The points equidistant from the $y$ axis and one of the points will be a parabola; its equation is easy to find. Now you have to find the intersection points between two parabolas, which will involve solving a quadratic equation.

HINT: Use circle property that angle in alternate segment equals angle between tangent and chord. Let given points be A,B, tangent point on y-axis be T ( 0,yT) and origin O(0,0). Find vectors AT,AB, TO and TB. Since $cos( OTB) = cos (TAB)$ find yT.

The Two points must be on the same side with respect to y axis. So I will take other example and that is when those two points are (1,3) and (2,4). Find the equations of all possible circles.

Step 1) Keep in mind that when a circle touch y axis, its radius is of the same value as its center's x-coordinate. (r=absolute (X center) ).

Step 2) Therefore Equation will be of the form (X-a)^2+(Y-b)^2=a^2 and we have to find a and b

Step 3) Substitute the two points int the equation:

(1-a)^2+(3-b)^2=a^2 and (2-a)^2+(4-b)^2=a^2 Solve the system. we get (a,b)=(1,4) or (5,0)