Statement I : The locus of the center of a variable circle touching two circles ${(x - 1)}^{2} + {(y - 2)}^{2} = 25$ and ${(x - 2)}^{2} + {(y - 1)}^{2} = 16$ is an ellipse.
Statement II : If a circle $S_{2} = 0$ lies completely inside the circle $S_{1} = 0$ , then the locus of the centre of a variable circle $S = 0$ that touches both the circles is an ellipse.

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Both statement I and II are true but statement II is the correct explanation of statement 1.

Both statement I and II are true but statement II is not the correct explanation of statement 1.

Statement I is true and statement II is false.

Statement I is false sand statement II is true.

answer is D.

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Detailed Solution

Let $C_{1}$ and $C_{2}$ be the centers and $r_{1}$ and $r_{2}$ be the radii of the two circles, respectively. Let $S_{1} = 0$ lies completely inside the circle. $S_{2} = 0$ . Let C and r be the center and radius of the variable circle, respectively. Then,
$C C_{2} = r_{2} - r a n d C_{1} C = r_{1} + r$
$C_{1} C + C_{2} C = r_{1} + r_{2} (c o n s t a n t)$
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Therefore, the locus of C is an ellipse.
Hence, $S_{2}$ is true.
Statement I is false (two circles are intersecting).