Question
Let $C_{1}$ and $C_{2}$ be two circles with $C_{2}$ lying inside $C_{1}$. A circle $C$ lying inside $C_{1}$ touches $C_{1}$ internally and $C_{2}$ externally. Identify the locus of the centre of $C$.
Step 1
Step 1: Let's denote the equations of the circles $C_{1}$ and $C_{2}$ as follows: $C_{1}$: $x^{2} + y^{2} = r_{1}^{2}$, $C_{2}$: $(x-a)^{2} + (y-b)^{2} = r_{2}^{2}$. Show more…
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Key Concepts
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Locus and Constructions
Locus
Assertion: The locus of the centres of circles passing through the origin and cutting the circle $x^{2}+y^{2}+6 x-$ $4 y+2=0$ orthogonally is $3 x-2 y+1=0$. Reason: The two circles $x^{2}+y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0$ and $x^{2}+y^{2}+2 g_{2} x+2 f_{2} y+c_{2}=0$ cut each other orthogonally if $2 g_{1} g_{2}+2 f f_{2}=c_{1}+c_{2}$
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