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(B) Let the centre of the circle $C$ be $(h, k)$. Since the circle touches the $X$-axis,the radius $r = |k|$.Given the centre lies on $y=x+1$,we have

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$x^2+y^2-26x-20y+19=0$

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(B) Let the centre of the circle $C$ be $(h, k)$. Since the circle touches the $X$-axis,the radius $r = |k|$.Given the centre lies on $y=x+1$,we have $k = h+1$.The equation of the circle is $(x-h)^2 + (y-k)^2 = k^2$.This circle makes an intercept of $2$ units on the $Y$-axis. Setting $x=0$,we get $h^2 + (y-k)^2 = k^2$,which simplifies to $y^2 - 2ky + h^2 = 0$.The length of the intercept is $|y_1 - y_2| = 2\sqrt{k^2 - h^2} = 2$.Thus,$k^2 - h^2 = 1$.Substituting $k = h+1$,we get $(h+1)^2 - h^2 = 1$,which implies $h^2 + 2h + 1 - h^2 = 1$,so $2h = 0$,giving $h=0$.Then $k = 0+1 = 1$.The centre of circle $C$ is $(0, 1)$.We need to find a circle passing through $(0, 1)$.Checking option $A$: $0^2 + 1^2 - 2(0) - 4(1) + 1 = 1 - 4 + 1 = -2 
eq 0$.Checking option $D$: $0^2 + 1^2 + 2(0) - 4(1) + 1 = 1 - 4 + 1 = -2 
eq 0$.Checking option $B$: $0^2 + 1^2 - 26(0) - 20(1) + 19 = 1 - 20 + 19 = 0$.Thus,the circle $x^2+y^2-26x-20y+19=0$ passes through $(0, 1)$.

$A$ circle $C$ touches the $X$-axis and makes an intercept of length $2$ units on the $Y$-axis. If the centre of this circle lies on the line $y=x+1$,then which of the following is a circle passing through the centre of the circle $C$?

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