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Find the center of the smallest circle w...

Find the center of the smallest circle which cuts circles `x^2+y^2=1` and `x^2+y^2+8x+8y-33=0` orthogonally.

Text Solution

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Let the equation of the required circle be `x^(2)+y^(2)+2gx+2fy+c=0`.
This circle cuts the given two circles othogonally.
`:. 2g(0)+2f(0)= -1+c,`
`:. c=1`
and `2(g)(4)+2(f)(4)= -33+c= -32`
`:. G+f= -4`
`:. `Radius of the circle `sqrt(g^(2)+f^(2)-c)`
`=sqrt(g^(2)+(g+4)^(2)-1) `
`=sqrt(2g^(2)+8g+15)`
`=sqrt(2(g+2)^(2)+7)`
Radius is minimum if `g+2=0` or `g =-2`.
`:. f= -1`
Hence, equation of the circle is `x^(2)+y^(2)-4x-4y+1=0`.
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