Question

The number of such points `(a+1,sqrt3a)`, where a is any integer, lying inside the region bounded by the circles `x^2 +y^2-2x-3=0` and `x^2+y^2-2x-15=0`, is

Answer 1

We have circles `(x-1)^(2)+y^(2)=4` (1)
and `(x-1)^(2)+y^(2)=16` (2)
Circles are concentric having centre at C(1,0) and radii 2 and 4.

Variable point `(a+1,sqrt(3)a)` lies on the line `y=sqrt(3)(x-1)` , which passes through centre of the circles having inclination `60^(@)`.
Line meets circles at `P(1+2cos 60^(@),2 sin 60^(@))`and `Q(1+4 cos 60^(@), 4 sin 60^(@))`.
Thus, `P-= (2,sqrt(3))` and `Q -= (3,2sqrt(3))`
`:. ain(2,3)`