Binomial Theorem

Question
CBSEENMA11015085

Find the co-ordinates of the centre and the radius of the circle
straight x squared plus straight y squared minus 2 ax space cosθ space minus space 2 ay space sinθ space minus space 8 straight a squared equals 0

Solution

The given equation is : straight x squared plus straight y squared minus 2 ax space cos space straight theta space minus space 2 ay space sinθ space minus space 8 straight a squared equals 0                   ...(i)
It can be written as
         space space left parenthesis straight x squared minus 2 ax space cosθ right parenthesis space plus space left parenthesis straight y squared minus 2 ay space sinθ right parenthesis space equals space 8 straight a squared
Completing the squares in each bracket, we haveleft parenthesis straight x squared minus 2 ax space cosθ space plus space straight a squared space cos squared straight theta right parenthesis space plus space left parenthesis straight y squared minus 2 ay space sinθ space plus space straight a squared sin squared straight theta right parenthesis space equals space 8 straight a squared plus straight a squared cos squared straight theta plus straight a squared sin squared straight theta
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or    left parenthesis straight x minus straight a space cosθ right parenthesis squared plus left parenthesis straight y minus straight a space sinθ right parenthesis squared space equals space 9 straight a squared space equals space left parenthesis 3 straight a right parenthesis squared
Comparing it with the standard form left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma we  get
                                  straight h equals straight a space cosθ comma space straight k equals straight a space sinθ comma space straight r equals 3 straight a
Hence, the centre of the circle is left parenthesis straight a space cosθ comma space straight a space sinθ right parenthesis and its radius is 3a.

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