If the circles x2+y2+6x+8y+16=0 and x2+y2+2(3−3)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 :

C1(−3,−4),C2(−3+3,−4+6),r1=3,r2=34+kC1C2=r1−r2⇒k=2 ∴r2=6∴P(α,β)=(−3−3,−4−6)∴(α+3)2+(β+6)2=9+16=25.

If the circles x2+y2+6x+8y+16=0 and x2+y2+2(3−3)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 :

C1(−3,−4),C2(−3+3,−4+6),r1=3,r2=34+kC1C2=r1−r2⇒k=2 ∴r2=6∴P(α,β)=(−3−3,−4−6)∴(α+3)2+(β+6)2=9+16=25.

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If the circles $x^{2} + y^{2} + 6 x + 8 y + 16 = 0$ and $x^{2} + y^{2} + 2 (3 - \sqrt{3}) x + 2 (4 - \sqrt{6}) y = k + 6 \sqrt{3} + 8 \sqrt{6}, k > 0$ touch internally at the point $P (α, β)$
$then (α + \sqrt{3})^{2} + (β + \sqrt{6})^{2}$ is :

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Detailed Solution

$\begin{array}{l} C_{1} (- 3, - 4), C_{2} (- 3 + \sqrt{3}, - 4 + \sqrt{6}), \\ r_{1} = 3, r_{2} = \sqrt{34 + k} \\ C_{1} C_{2} = |r_{1} - r_{2}| \Rightarrow k = 2 \\ ∴ r_{2} = 6 \\ ∴ P (α, β) = (- 3 - \sqrt{3}, - 4 - \sqrt{6}) \\ ∴ (α + \sqrt{3})^{2} + (β + \sqrt{6})^{2} = 9 + 16 = 25 . \end{array}$