Given the circle C: X & # 178; + Y & # 178; - 2x + 4y-4 = 0, ask whether there is a straight line L with a slope of 1, so that l is cut by the circle C If there is a circle with the diameter of string AB passing through the origin, write the equation of line L; if there is no such a circle, explain the reason

Given the circle C: X & # 178; + Y & # 178; - 2x + 4y-4 = 0, ask whether there is a straight line L with a slope of 1, so that l is cut by the circle C If there is a circle with the diameter of string AB passing through the origin, write the equation of line L; if there is no such a circle, explain the reason

Suppose there is a straight line L: y = x + λ
x-y+λ=0
simultaneous
｛x²+y²-2x+4y-4=0
｛y=x+λ
x²+(x+λ)²-2x+4(x+λ)-4=0
x²+(x²+2λx+λ²)-2x+4(x+λ)-4=0
2x²+2(λ+1)x+(λ²+4λ-4)=0
x1+x2=-(λ+1)
x1x2=(λ²+4λ-4)/2
According to the meaning of the title:
OA⊥OB
x1x2+y1y2=0
x1x2+(x1+λ)(x2+λ)=0
2x1x2+λ(x1+x2)+λ²=0
(λ²+4λ-4)-λ(λ+1)+λ²=0
λ²+3λ-4=0
λ1=1
λ2=-4
L1:y=x+1
L2:y=x-4
If you find that the answer is wrong, it is wrong calculation. The method is right and can be changed