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Qtn : 3

Find out the distance between the following pairs of points:

  1. (a cos α , a sin α) and ( a cos β, a sin β )
  2. (a m1 2, 2 am1) and a m1 2, 2 am1
  3. (6a, a/2) and (-4a,5a/2)

Asked by miniprasad | 20th Oct, 2018, 05:02: PM

Expert Answer:

  1. (a cos α , a sin α) and ( a cos β, a sin β )

Consider A(a cos α , a sin α) and B ( a cos β, a sin β )

Distance between A and B = begin mathsize 16px style square root of open parentheses acosα minus acos space straight beta close parentheses squared plus open parentheses asin space straight alpha space minus space asinβ close parentheses squared end root end style

begin mathsize 16px style equals square root of straight a squared cos squared straight alpha minus 2 straight a squared cosαcosβ plus straight a squared cos squared straight beta plus straight a squared sin squared straight alpha minus 2 straight a squared sinαsinβ plus straight a squared sin squared straight beta end root
equals square root of straight a squared cos squared straight alpha plus straight a squared sin squared straight alpha minus 2 straight a squared cosαcosβ minus 2 straight a squared sinαsinβ plus straight a squared sin squared straight beta plus straight a squared cos squared straight beta end root
equals square root of straight a squared open parentheses cos squared straight alpha plus sin squared straight alpha close parentheses minus 2 straight a squared open parentheses cosαcosβ plus sinαsinβ close parentheses plus straight a squared open parentheses sin squared straight beta plus cos squared straight beta close parentheses end root
equals square root of straight a squared minus 2 straight a squared open parentheses cosαcosβ plus sinαsinβ close parentheses plus straight a squared end root
equals square root of 2 straight a squared minus 2 straight a squared cos open parentheses straight alpha minus straight beta close parentheses end root
equals square root of 2 straight a squared open parentheses 1 minus cos open parentheses straight alpha minus straight beta close parentheses close parentheses end root
equals straight a square root of 2 cross times 2 sin squared open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses end root
equals 2 asin open parentheses fraction numerator straight alpha minus straight beta over denominator 2 end fraction close parentheses end style

Answered by Sneha shidid | 22nd Oct, 2018, 01:14: PM