Integer Answer Type Questions: Circle - JEE Advanced

Q.1. The centres of two circles C₁ and C₂ 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 C₁ and C₂ and 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 C₁C₂ 
∴ CP ⊥ C₁C₂
Also PM ⊥ CC₁
Now ΔPMC₁ ~ΔCPC₁ (by AA similarity)

 r  + 1 = 9 ⇒ r =  8.

Q.2. The straight line 2x - 3y = 1 divides the circular region x² + y² ≤ 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 x² + y² < 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.