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Q.

If the circles x2 + y2 + 6x + 8y + 16 = 0 and x2+y2+2(33)x+x+2(46)y=k+63+86,k>0, touch internally at the point P(α,β), then (α+3)2+(β+6)2 is equal to __________

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answer is 25.

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Detailed Solution

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x2 + y2 + 6x + 8y + 16 = 0 has centre (-3,-4) and radius=3

x2+y2+2(33)x+2(46)y=k+63+86,k>0 has centre (33,64) and radius k+34

These two circles touch internally

C1C2=r1-r2 3+6=k+343 k=2k>0  Equation of common tangent to two circles is  23x+26y+16+63+86+k=0 substitute k=2 x+2y+3+42+33=0 α,β are foot of perpendicular from3,4  to line i then  α+31=β+42=(342+3+42+33)1+2α+3=β+42=3(α+3)2=9 and (β+6)2=16(α+3)2+(β+6)2=25 

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If the circles x2 + y2 + 6x + 8y + 16 = 0 and x2+y2+2(3−3)x+x+2(4−6)y=k+63+86,k>0, touch internally at the point P(α,β), then (α+3)2+(β+6)2 is equal to __________