Three circle C1, C2 and C3 with radii r1, r2 and r3 (where r1 < r2 <… — Geometry MCQ

Aptitude · Q20

Geometry

Competitive Exams · Aptitude · question 20

Q20

Three circle C₁, C₂ and C₃ with radii r₁, r₂ and r₃ (where r₁ < r₂ < r₃) are placed as shown in the given figure. What is the value of r₂?
Geometry mcq question image

Geometry mcq question image

A.

r_{1} r_{3}

Answer

B.

\frac{r _{1} + r _{3}}{2}

C.

\frac{2 r _{1} r _{2}}{r _{1} + r _{2}}

D.

r_{1} + r_{3}

Answer: Option A

Solution

Answer: Option A

Solution:

Geometry mcq question image

A B = 4 r_{1} . r_{2} B C = 4 r_{2} . r_{3} A C = (r_{1} + 2 r_{2} + r_{3})^{2} - (r_{3} - r_{1})^{2} A B + B C = A C 4 r_{1} . r_{2} + 4 r_{2} . r_{3} = (r_{1} + 2 r_{2} + r_{3})^{2} - (r_{3} - r_{1})^{2} Squaring both sides \Rightarrow 4 r_{1} . r_{2} + 4 r_{2} . r_{3} + 2 16 r_{1} . r_{2}^{2} . r_{3} = (r_{1} + 2 r_{2} + r_{3})^{2} - (r_{3} - r_{1})^{2} \Rightarrow 4 r_{1} . r_{2} + 4 r_{2} . r_{3} + 8 r_{2} r_{1} . r_{3} = r_{1}^{2} + 4 r_{2}^{2} + r_{3}^{2} + 4 r_{1} . r_{2} + 4 r_{2} . r_{3} + 2 r_{1} . r_{3} - r_{3}^{2} - r_{1}^{2} + 2 r_{1} . r_{3} \Rightarrow 8 r_{2} r_{1} . r_{3} = 4 r_{2}^{2} + 4 r_{1} . r_{3} \Rightarrow 2 r_{2} r_{1} . r_{3} = r_{2}^{2} + r_{1} . r_{3} Squaring both sides \Rightarrow 4 r_{2}^{2} . r_{1} . r_{3} = r_{2}^{4} + r_{1}^{2} . r_{3}^{2} + 2 r_{2}^{2} . r_{1} . r_{3} \Rightarrow (r_{2}^{2} - r_{1} . r_{3})^{2} = 0 \Rightarrow r_{2}^{2} = r_{1} . r_{3} \Rightarrow r_{2} = r_{1} . r_{3}

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