IIT-JEE 2009 · Paper 2 · Mathematics Q36

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MathematicsInteger+4 / -1Question 36 of 57

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$ , then the radius of the circle $C$ is

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Answer and worked solution

Official answer: 8

Set-up. $C_1(-3,0)$ , $C_2(3,0)$ , $P=(0,0)$ . A line through P at distance 1 from $C_1$ is also at distance 1 from $C_2$ (symmetry): $y=x\tan\alpha$ with $\sin\alpha=\frac13$ , $\cos\alpha=\frac{2\sqrt2}3$ .

Circle C. By symmetry its centre is $(0,h)$ . Tangent to the line: $r=h\cos\alpha=\frac{2\sqrt2}3h$ . Touching $C_1$ externally: $\sqrt{9+h^2}=r+1$ .

Solve. With $h=\frac{3r}{2\sqrt2}$ : $9+\frac98r^2=r^2+2r+1$ , i.e. $r^2-16r+64=0$ , so $r=8$ .

Answer: 8

Topics:Tangents and normals Equation of a circle

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IIT-JEE 2009 · Paper 2, Mathematics question 36