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Q.1. 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 segement 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
(2009)
Ans. (8)
Sol. Let r be the radius of required circle.
Clearly, in ΔC_{1}CC_{2} , C_{1}C = C_{2}C =r+ 1
and P is mid point of C_{1}C_{2 }
∴ CP ⊥ C_{1}C_{2}
Also PM ⊥ CC_{1}
Now ΔPMC_{1} ~ΔCPC_{1} (by AA similarity)
r + 1 = 9 ⇒ r = 8.
Q.2. The straight line 2x – 3y = 1 divides the circular region x^{2} + y^{2} ≤ 6 into two parts.
If then the number ofpoints (s) in S lying inside the smaller part is (2011)
Ans. (2)
Sol.
The smaller region of circle is the region given by x^{2} + y^{2} < 6 ...(1)
and 2x – 3y > 1 ...(2)
We observe that only two points and
satisfy both the inequations (1) and (2)
∴ 2 points in S lie inside the smaller part.
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