# Find the radius of 2 circles

## 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*

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

Why don't you take the known information and put it on a graph. It's a start.
Without graph it should be

pasmith
Homework Helper
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 $c$.

• Lifeforbetter
ehild
Homework Helper
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.)

View attachment 246567

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• 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.)

View attachment 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.

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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?

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