Find the equation of a circle with radius 5 u | vedclass

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(A) The equation of the given circle is $x^2+y^2-2x-4y-20=0$.Rewriting in standard form: $(x-1)^2+(y-2)^2=25$.The center is $C_1(1,2)$ and the radius

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$x^2+y^2-18x-16y+120=0$

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(A) The equation of the given circle is $x^2+y^2-2x-4y-20=0$.Rewriting in standard form: $(x-1)^2+(y-2)^2=25$.The center is $C_1(1,2)$ and the radius is $r_1=5$.Let the center of the required circle be $C_2(h,k)$ and its radius be $r_2=5$.Since the circles touch at $P(5,5)$,the point $P$ lies on the line segment $C_1C_2$.Because $r_1=r_2=5$,$P$ is the midpoint of $C_1C_2$.Using the midpoint formula: $\frac{1+h}{2}=5 \Rightarrow h=9$ and $\frac{2+k}{2}=5 \Rightarrow k=8$.Thus,the center $C_2$ is $(9,8)$.The equation of the second circle is $(x-9)^2+(y-8)^2=5^2$.Expanding this: $x^2-18x+81+y^2-16y+64=25$.Simplifying: $x^2+y^2-18x-16y+120=0$.

Find the equation of a circle with radius $5$ units and touching the circle $x^2+y^2-2x-4y-20=0$ at the point $(5,5)$.

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