A circle C1 with radius 5 touches x-axis and another circle C2 with radius 4 touches y-axis. These two circles touch each other externally at P(α,β) (α,β,≥0). let the locus of the point P be the curve S. Two perpendicular tangents are drawn to the curve S from Q(a,b) then the value of (a−4)2+(b−5)2= ________

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A circle $C_{1}$ with radius 5 touches x-axis and another circle $C_{2}$ with radius 4 touches y-axis. These two circles touch each other externally at $P (α, β) (α, β, \geq 0) .$ let the locus of the point P be the curve S. Two perpendicular tangents are drawn to the curve S from $Q (a, b)$ then the value of ${(a - 4)}^{2} + {(b - 5)}^{2} =$ _________

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Detailed Solution

C (h, k) divides AB into the ratio 5 : 4.
$∴ (h, k) = (\frac{4 α + 20}{9}, \frac{5 β + 20}{9})$
$A B = 9 \Rightarrow {(α - 4)}^{2} + {(β - 5)}^{2} = 9^{2}$
$\Rightarrow \frac{{(h - 4)}^{2}}{4^{2}} + \frac{{(k - 5)}^{4}}{5^{2}} = 1$
Locus of point of contact is $\frac{{(x - 4)}^{2}}{16} + \frac{{(y - 5)}^{2}}{25} = 1$