Find circle given diameter and string function

Question

I have a circle that the center goes through $ y = 2x$ and $R=2$.

I need to find the circle equation if it's given that $y = x+3$ cuts the circle in a string of length $\sqrt{8}$.

I drew all this information in GeoGebra $5$ and I think it would look somehow like this:

Besides all the information I created in the drawing, I don't know what more can I do.

Answer 1

Hint

Let $D\; :\; x-y+3=0.$

the distance from the center $A(a,2a)$ to $D$ is

$$\frac{|a-2a+3|}{\sqrt{1^2+(-1)^2}}$$

$=\sqrt{2}\;\;$ by Phytagoras.

thus

$$|3-a|=2$$

which gives $\;A(1,2)$ or $A(5,10)$

and the equations

$$(x-1)^2+(y-2)^2=4$$

or

$$(x-5)^2+(y-10)^2=4.$$

Answer 2

Check that distance of $(a,2a)$ from $y=x+3$ should be $\sqrt 2$

Now your turn.