(1) The ratio of the arc length of two arcs divided by the X axis is 3: 1.
Explain that the central angle is 360× 1/4 = 90.
|b|= (root number 2)r/2
(2) The chord length obtained by cutting the Y axis of circle C is 2.
Description |a|= root symbol (r? - 1)
(3) The distance from the center of the circle C to the straight line L: X-2Y = 0 is the radical number 5 of 5.
According to the distance formula from point to line: |a-2b|/ (root number 5)= (root number 5)/5.
|a-2b|= 1
From (1)(2)(3), we get
R= root number 2
|a|= 1
|b|= 1
According to |a-2b|= 1, we can know that a= 1, b= 1 or a=- 1, and b=- 1.
So the expression of the circle is
(x- 1)？ +(y- 1)？ =2
perhaps
(x+ 1)？ +(y+ 1)？ =2
Reference:
Let the center of the circle be P(a, b) and the radius be r,
Then the distances from P to X axis and Y axis are |b| and |a| respectively.
It is known that the central angle of the lower arc obtained by cutting the X axis by the circle P is 90 degrees, and the chord length obtained by cutting the X axis by the circle P is (root number 2)*r, so
r^2=2b
The chord length of the circle p tangent to the Y axis is 2, so there is
r^2=a^2+ 1
In order to obtain
2b^2-a^2= 1
And the distance from P(a, b) to the straight line x-2y=0 is
D=|a-2b|/ root 5
-& gt； 5d^2=a^2+4b^2-4ab>； =a^2+4b^2-2(a^2+b^2)=2b^2-a^2= 1
When a=b, the above equal sign holds.
At this time, 5d 2 = 1, so d takes the minimum value.
Therefore, {a = b, 2b 2-a2 =1}
-& gt； A=b= 1, or a=b=- 1.
Since r 2 = 2b 2, r= radical number 2.
Therefore, the equation for finding a circle is:
(x- 1)^2+(y- 1)^2=2，
Or (x+ 1) 2+(y+ 1) 2 = 2.