Principle Of Mathematical Induction

Question
CBSEENMA11015498

ABCD is a trapezium such that AB and CD are parallel and BC⊥CD. If ∠ADB = θ, BC = p and CD = q,then AB is equal to

  • fraction numerator left parenthesis straight p squared plus straight q squared right parenthesis space sin space straight theta over denominator straight p space cos space straight theta space plus space straight q space sin space straight theta end fraction
  • fraction numerator straight p squared space plus straight q squared space cos space straight theta over denominator straight p squared space cos space straight theta space plus space straight q space sin space straight theta end fraction
  • fraction numerator straight p squared plus straight q squared space cos space straight theta over denominator straight p squared space cos space straight theta space plus straight q space sin space straight theta end fraction
  • fraction numerator left parenthesis straight p squared plus straight q squared right parenthesis over denominator left parenthesis straight p space cos space straight theta space plus space straight q space sin space straight theta right parenthesis squared end fraction

Solution

A.

fraction numerator left parenthesis straight p squared plus straight q squared right parenthesis space sin space straight theta over denominator straight p space cos space straight theta space plus space straight q space sin space straight theta end fraction
Using space sine space rule space in space triangle space ABD
fraction numerator AB over denominator sin space straight theta end fraction space equals fraction numerator BD over denominator sin space left parenthesis straight theta space plus straight alpha right parenthesis end fraction
rightwards double arrow space AB space equals space fraction numerator square root of straight p squared plus straight q squared end root space sin space straight theta over denominator sin space straight theta space cos space straight alpha space plus space cos space straight theta space sinα end fraction space
equals space fraction numerator square root of straight p squared plus straight q squared end root sin space straight theta over denominator begin display style fraction numerator sin space straight theta. straight q over denominator square root of straight p squared plus straight q squared end root end fraction end style space plus begin display style fraction numerator cosθ. straight P over denominator square root of straight p squared plus straight q squared end root end fraction end style end fraction
rightwards double arrow AB space equals space fraction numerator left parenthesis straight p squared plus straight q squared right parenthesis space sin space straight theta over denominator left parenthesis straight p space cos space straight theta space plus space straight q space sin space straight theta right parenthesis end fraction

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