Two straight lines AB and AC include an angle A circle is drawn in this angle which touches both the...........

Quantitative Aptitude (CDS) Mensuration 2D

Two straight lines AB and AC include an angle. A circle is drawn in this angle which touches both these lines. One more circle is drawn which touches both these lines as well as the previous circle. If the area of the bigger circle is 9 times the area of the smaller circle, then what must be the angle A?

45°

60°

75°

90°

छोटे Courses बड़े Results

Master Mensuration 2D

Prepare through Micro-courses

Try Now

Explanation:

Let the radius of bigger circle be x and radius of the smaller circle be y.
Then the angle made by direct common tangents when two circles of radius x and y touch externally is given by
$\Theta =2 \;sin^{-1}\frac{x-y}{x+y}$
Given that area of the bigger circle = 9 ×(area of the smaller circle)
=> $\pi$ x²=9 $\pi$ y²=>x²=9y²=>x=3y
Let us consider Angle BAC = θ
$\Theta =2 \;sin^{-1}\frac{x-y}{x+y}=2 \;sin^{-1}\frac{3y-y}{3y+y}=2 \;sin^{-1}\frac{2y}{4y}=2 \;sin^{-1}\frac{1}{2}$
=2*30^o=60^o