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Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q.1. The centres of two circles C1 and Ceach 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 C1 and Cand C be a circle touching circles C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is
      (2009)

Ans. (8)

Sol.   Let r be the radius of required circle.
Clearly, in ΔC1CC2 , C1C = C2C =r+ 1
and P is mid point of C1C
∴ CP ⊥ C1C2
Also PM ⊥ CC1
Now ΔPMC1 ~ΔCPC1 (by AA similarity)

Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE r  + 1 = 9 ⇒ r =  8.

 

Q.2. The straight line 2x – 3y = 1 divides the circular region x2 + y2 ≤ 6 into two parts. 

If Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 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 x2 + y2 < 6 ...(1)
and 2x – 3y > 1 ...(2)

Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

We observe that only two points Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and  Integer Answer Type Questions: Circle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

satisfy both the inequations (1) and (2)

∴ 2 points in S lie inside the smaller part.

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