Prove that the circle x^2 +y^2 - 6x-2y + 9 = 0
i) touches the X axis
ii) lies entirely inside the circle x^2 + +y^ = 18

Dear Student,
Please find below the solution to the asked query:

We haveS1:x2+y2-6x-2y+9=0For x-axis, y=0x2-6x+9=0x-32=0x=3,3As quadratic equation has real and equal roots, hence circle touches x-axis.Centre of circle S1 is C1-g,-f=C13,1Radius=r1=g2+f2-c=9+1-9=1Centre of circle S2: x2+y2=18=322  is C20,0 and radius is r2=32C1C2=3-02+1-02=10r1-r2=1-32=32-1As C1C2<r1-r2, henceS1 lies entirely inside S2. 

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