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Question

The line x cos alpha + y sin alpha = p will be a tangent to the conic fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction+ fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction= 1, if-

  1. p2 = a2 sin2 alpha+ b2 cos2alpha    
  2. p2 = a2 + b2    
  3. p2 = b2 sin2alpha + a2 cos2alpha    
  4. None of these    

hintHint:

solve the equations and make the discriminant value 0 to get the required value of p

The correct answer is: p2 = b2 sin2alpha + a2 cos2alpha


    p2=a2 cos2 α + b2sin2 α

    given equation of tangent : xcos α + y sin α =p
    using the value of y from this equation into the equation of ellipse, we get
    x2/a2+((p-x cos  α )/sin α )2/b2=1
    => (a2 cos2  α +b2sin2  α )x2-(2pa2cos α )x +p2a2-a2b2sin2  α =0
    For the line to be a tangent, D=0
    => b2=4ac
    => 4a4 b2 sin2 α cos2 α=4p2 a2 b2 sin2 α−4a2 b4 sin4 α
    => p2=a2 cos2 α + b2sin2 α

    the tangent touches the curve at a single point and hence, has a single real solution to the equation.

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