Maths-
General
Easy

Question

If ϕ is the angle between the diameter through any point on a standard ellipse and the normal at the point, then the greatest value of tan ϕ is–

  1. fraction numerator 2 a b over denominator a to the power of 2 end exponent plus b to the power of 2 end exponent end fraction    
  2. fraction numerator a to the power of 2 end exponent plus b to the power of 2 end exponent over denominator a b end fraction    
  3. fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction    
  4. fraction numerator b to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction    

The correct answer is: fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction


    Any point P on ellipse is (a cos theta, b sin theta)
     Equation of the diameter CP is y = open parentheses fraction numerator b over denominator a end fraction tan invisible function application theta close parenthesesx
    The normal to ellipse at P is
    ax sec theta – by cosec theta = a2e2
    Slopes of the lines CP and the normal GP are fraction numerator b over denominator a end fractiontan theta andfraction numerator a over denominator b end fractiontan theta

     tan ϕ = fraction numerator fraction numerator a over denominator b end fraction tan invisible function application theta minus fraction numerator b over denominator a end fraction tan invisible function application theta over denominator 1 plus fraction numerator a over denominator b end fraction tan invisible function application theta. fraction numerator b over denominator a end fraction tan invisible function application theta end fraction=fraction numerator tan invisible function application theta over denominator sec to the power of 2 end exponent invisible function application theta end fraction
    =fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator a b end fractionsin  cos  = fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fractionsin 2
     The greatest value of tan ϕ = fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction.1 = fraction numerator a to the power of 2 end exponent minus b to the power of 2 end exponent over denominator 2 a b end fraction .

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