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Question : AB is the common tangent to both circles as shown in the given figure. What is the distance between the centres of the circles?

Option 1: 20 cm

Option 2: 15 cm

Option 3: 10 cm

Option 4: 30 cm


Team Careers360 13th Jan, 2024
Answer (1)
Team Careers360 23rd Jan, 2024

Correct Answer: 30 cm


Solution :
Given that AB is the common tangent to both circles.
In the circle with centre C, $\angle$CAE = 90°
In the circle with centre D, $\angle$DBA = 90°
and $\angle$AEC = $\angle$BED (vertically opposite angle)
So, $\triangle$CAE $\sim$ $\triangle$DBC
⇒ $\frac{CA}{AE}=\frac{DB}{BE}$
⇒ $\frac{4}{3}=\frac{DB}{15}$
⇒ DB = $\frac{4×15}{3}$ = 20 cm
In $\triangle$CAE,
⇒ CE2 = AC2 + AE2
⇒ CE2 = 42 + 32 = 52
⇒ CE = 5 cm
In $\triangle$DBE,
⇒ DE2 = DB2 + BE2
⇒ DE2 = 202 + 152 = 252
⇒ DE = 25 cm
$\therefore$ The distance between the centres of the circle = CE + DE = 5 + 25 = 30 cm
Hence, the correct answer is 30 cm.

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