Maths-
General
Easy

Question

PQ and QR are two focal chords of an ellipse and the eccentric angles of P,Q,R and
2 alpha, 2 beta comma blank 2 gammarespectively then tan beta tan gamma is equal to -

  1. cot alpha    
  2. cot2alpha    
  3. 2 cot alpha    
  4. None of these    

The correct answer is: cot2alpha


    fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1
    P (a cos 2 alpha, b sin 2 alpha), Q (a cos 2 beta , b sin 2 beta)
    R (a cos 2 gamma, b sin 2 gamma)
    chord's PQ equation
    fraction numerator x over denominator a end fractioncos (alpha + beta) + fraction numerator y over denominator b end fractionsin (alpha + beta) = cos (alphabeta)
    PQ passes through the focus (ae, 0)
    e = fraction numerator cos invisible function application left parenthesis alpha minus beta right parenthesis over denominator cos invisible function application left parenthesis alpha plus beta right parenthesis end fraction
    PR passes through the focus (– ae, 0) the
    – e = fraction numerator cos invisible function application left parenthesis alpha minus gamma right parenthesis over denominator cos invisible function application left parenthesis alpha plus gamma right parenthesis end fraction
    fraction numerator cos invisible function application left parenthesis alpha minus beta right parenthesis over denominator cos invisible function application left parenthesis alpha plus beta right parenthesis end fraction = – fraction numerator cos invisible function application left parenthesis alpha minus gamma right parenthesis over denominator cos invisible function application left parenthesis alpha plus gamma right parenthesis end fraction
    Apply componendo and dividendo, we get
    fraction numerator cos invisible function application left parenthesis alpha plus beta right parenthesis plus cos invisible function application left parenthesis alpha minus beta right parenthesis over denominator cos invisible function application left parenthesis alpha plus beta right parenthesis minus cos invisible function application left parenthesis alpha minus beta right parenthesis end fraction = fraction numerator cos invisible function application left parenthesis alpha plus gamma right parenthesis minus cos invisible function application left parenthesis alpha minus gamma right parenthesis over denominator cos invisible function application left parenthesis alpha plus gamma right parenthesis plus cos invisible function application left parenthesis alpha minus gamma right parenthesis end fraction
    fraction numerator 2 cos invisible function application alpha cos invisible function application beta over denominator 2 sin invisible function application alpha sin invisible function application beta end fraction = fraction numerator 2 sin invisible function application alpha sin invisible function application gamma over denominator 2 cos invisible function application alpha cos invisible function application gamma end fraction
    tan beta tan gamma = cot2 alpha

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