How to Calculate the Radii of Two Circles Passing Through a Common Point?

[Lifeforbetter]

Homework Statement

Two circles go through 2 points (1,3) (2,4)
Both circle touches y-axis. Find r1*r2
The mutiple of the radius of both circle

Relevant Equations

d = $\sqrt{(x2^2-x1^2) (y2^2 - y1^2) }$

Middle point of (1,3)(2,4) is (1.5, 3.5)
r1 to r2 passing through (1.5, 3.5)
I cannot grasp on what should i do to find r1 and r2 from the line

Without graph*

 

[magoo]

Why don't you take the known information and put it on a graph. It's a start.

 

[Lifeforbetter]

Why don't you take the known information and put it on a graph. It's a start.

Without graph it should be

 

[pasmith]

If the center of a circle of radius R is on the line $y = mx + c$ and the circle touches the y-axis, then the equation of that circle must be $$(x - R)^2 + (y - (mR + c))^2 = R^2.$$ If you know $(x,y)$ (a point on the circle) and $m$ and $c$ then this is a quadratic you can solve to find $R$.

Here you know two points on the circle, $(x_1,y_1) = (1,3)$ and $(x_2,y_2) = (2,4)$. There are at least two ways of using this information to find $m$ and $c$.

 

[ehild]

Without graph it should be

Who will know if you made a sketch for yourself? You can see the problem better from a graph. Solution without graph means that you can not read the results from the graph.
Both circles go through both points P1(1;3) and P2(2;4). And both circles touch the y axis. The centers of the circles are O1(x₁;y₁) and O2(x₂;y₂), there radii are r1, r2. See the "forbidden" graph (not in scale).

How is the x position of the center of a circle related to the radius if the circle touches the y axis?
(The y-axis is tangent to the circle, the radius drawn to the tangent point is perpendicular to the tangent line.)

https://www.physicsforums.com/attachments/246567

 

[Lifeforbetter]

How is the x position of the center of a circle related to the radius if the circle touches the y axis?
(The y-axis is tangent to the circle, the radius drawn to the tangent point is perpendicular to the tangent line.)

https://www.physicsforums.com/attachments/246567

If a1, b1 were the center point of circle 1. Then a1 = r1, then b1 = mr1 + c
Just what @pasmith says right?
The graph does help too.

 

[Lifeforbetter]

If the center of a circle of radius R is on the line $y = mx + c$ and the circle touches the y-axis, then the equation of that circle must be $$(x - R)^2 + (y - (mR + c))^2 = R^2.$$ If you know $(x,y)$ (a point on the circle) and $m$ and $c$ then this is a quadratic you can solve to find $R$.

Here you know two points on the circle, $(x_1,y_1) = (1,3)$ and $(x_2,y_2) = (2,4)$. There are at least two ways of using this information to find $m$ and $c$.

y =mx + c
You mean the line go through middle point between P1 and P2? Which also go through r1 and r2 right?
That's m = -1 c = 5
Plug into
$$(x - R)^2 + (y - (mR + c))^2 = R^2.$$
Either using (1,3) or (2,4)
Give me r = 1 or r = 5 right?

 

[Lifeforbetter]

Why don't you take the known information and put it on a graph. It's a start.

Yes.