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Question

Two circles with radii apostrophe r subscript 1 superscript blank apostrophe text  and ' end text r subscript 2 apostrophe comma r subscript 1 greater than r subscript 2 greater or equal than 2, touch each other externally. If apostrophe theta apostrophe be the angle between the direct common tangents, then

  1. theta equals sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator r subscript 1 end subscript plus r subscript 2 end subscript over denominator r subscript 1 end subscript minus r subscript 2 end subscript end fraction close parentheses    
  2. theta equals 2 sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator r subscript 1 end subscript minus r subscript 2 end subscript over denominator r subscript 1 end subscript plus r subscript 2 end subscript end fraction close parentheses    
  3. q = 2cos to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 2 square root of r subscript 1 end subscript r subscript 2 end subscript end root over denominator r subscript 1 end subscript plus r subscript 2 end subscript end fraction close parentheses    
  4. none of these    

hintHint:


The correct answer is: theta equals 2 sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator r subscript 1 end subscript minus r subscript 2 end subscript over denominator r subscript 1 end subscript plus r subscript 2 end subscript end fraction close parentheses


    To find the angle between the direct common tangents.

    The line joining the point of intersection of two direct tangents with centers of circles bisect the angle.

    I n space t r i a n g l e space B C A comma
sin space theta over 2 equals fraction numerator C A over denominator C B end fraction equals fraction numerator r subscript 1 minus r subscript 2 over denominator r subscript 1 plus space r subscript 2 end fraction


    theta equals 2 sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator r subscript 1 end subscript minus r subscript 2 end subscript over denominator r subscript 1 end subscript plus r subscript 2 end subscript end fraction close parentheses

    Therefore, using tangent property of circles, theta equals 2 sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator r subscript 1 end subscript minus r subscript 2 end subscript over denominator r subscript 1 end subscript plus r subscript 2 end subscript end fraction close parentheses.

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