How to find the radii of these 2 circles given 2 known points

[Helly123]

Homework Statement

Homework Equations

y-y1 = m (x-x1) ---> line equation

$$ (x - a)^2 + (y-b)^2 = r^2 $$ ---> circle equation

The Attempt at a Solution

I tried to draw the triangles using, (1, 3) (2, 4) and (0, b)

(0, b) is the tangent point to y-axisand used those points for making perpendicular bisector, so I can find the center of the circle, and find the distance from that center to one of the points, which that distance is the radii.between (1, 3) (2, 4)

the mid point = 1.5 , 3.5 gradient = 1, the perpendicular m = -1

the line equation = y - 1.5 = -1(x - 3.5) ... (1)between (2,4) (0, b)

$$ mid point = (1, \frac {4+b)} {2} )

\\gradient = \frac {4-b}{2} \\the perpendicular gradient = \frac {-2}{4-b}

\\the line equation = y - \frac {(4+b)} {2} = \frac {-2}{(4-b)} (x - 1) ...(2) $$between (1,3) (0,b)

$$ \\mid point = (0.5, \frac {(3+b)} {2} )

\\gradient = \frac {(3-b)}{1} \\the perpendicular gradient = \frac {-1}{(3-b)}

\\the line equation = y - \frac {(3+b)} {2} = \frac {-1}{(3-b)} (x - 0.5) ...(3)

$$if I find the x, y then its the center

we know that the center (x, b) where b the same in b (0, b)

but i don't think its work.
can anyone give me another way to solve?

I find another solutions but I don't get what the person doing can anyone explain it?
https://www.algebra.com/algebra/homework/Circles/Circles.faq.question.1077722.html

 

[jbriggs444]

Because we are asked not to graph this, I would rule out most approaches based on geometry and go for something algebraic.

By inspection, the centers of both circles are at y=2. We do not know their respective x coordinates. So write down some equations.

Let $(x_1,y_1)$ be the center of the first circle and let $r_1$ be its radius.

The radius of the first circle is given by the distance from its center to the tangent point on the y axis:
$$r_1 = x_1$$
The radius of the first circle is also given by the distance from its center to point (1,3):
$$r_1^2 = (1-x_1)^2 + (3-y_1)^2$$
We already know that $y_1 = 2$. We already have that $r_1 = x_1$. Making those substitutions:

$$x_1^2 = (1-x_1)^2 + 1^2$$

Solve for $x_1$

Repeat for the second circle.

 

[Helly123]

Because we are asked not to graph this, I would rule out most approaches based on geometry and go for something algebraic.

By inspection, the centers of both circles are at y=2. We do not know their respective x coordinates. So write down some equations.

Let $(x_1,y_1)$ be the center of the first circle and let $r_1$ be its radius.

The radius of the first circle is given by the distance from its center to the tangent point on the y axis:
$$r_1 = x_1$$
The radius of the first circle is also given by the distance from its center to point (1,3):
$$r_1^2 = (1-x_1)^2 + (3-y_1)^2$$
We already know that $y_1 = 2$. We already have that $r_1 = x_1$. Making those substitutions:

$$x_1^2 = (1-x_1)^2 + 1^2$$

Solve for $x_1$

Repeat for the second circle.

how do you know the center y = 2? inspection of what?

 

[jbriggs444]

how do you know the center y = 2? inspection of what?

My mistake. I completely misread the problem somehow.

 

[haruspex]

but i don't think its work.

You seemed to be doing ok. It should get you there.

What do you not understand in the solution you linked?

 

[Helly123]

What do you not understand in the solution you linked?

From the first, finding center of upper circle , how it comes up (1.5 -P/root 2 , 3.5 + P/root 2) ?
And how to find radii? R^2 = (1.5 -P/root 2)^2 + (3.5 + P/root 2)^2 not the same answer as from the link

While my method i get imaginary x

 

[haruspex]

From the first, finding center of upper circle , how it comes up (1.5 -P/root 2 , 3.5 + P/root 2) ?
And how to find radii? R^2 = (1.5 -P/root 2)^2 + (3.5 + P/root 2)^2 not the same answer as from the link

While my method i get imaginary x

What is P?

I would have done this:
Centre of circle is at (R,b). This is distance R from each of the two given points...

 

[Helly123]

What is P?

I would have done this:
Centre of circle is at (R,b). This is distance R from each of the two given points...

So i used your method
$$ (R-1)^2 + (b-3)^2 = (R-2)^2 + (b-4)^2 $$
R + b = 5
b = 0 or 4
R = 5 or 1
So does it mean the center of circle A = R, b = 1, 4
And circle B = R, b = 5, 0
??