UPSC 2017 CDS II Math Paper

Let r and R be the radii of the smaller circle and bigger circle respectively.
∴πR2=9×πr2⇒

R2

r2

=9⇒

R

r

=3
Now,ΔOAP ~ ΔAO'AQ
∴

AO′

AO

=

R

r

⇒

AO+r+R

AO

=3
⇒AO+r+3r=3AO
⇒4r=2AO
⇒

r

AO

=

1

2

In right ΔOAP,
sin‌θ=

r

AO

=

1

2

⇒ θ = 30°
Remember:
If two tangents are drawn to a circle from an externalpoint, then tangents are equally inclined to the segment joining the centre to that point.
∴ ∠A=2×30°=60°