Maths-
General
Easy

Question

The tangent at any point on the ellipse 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 meets the tangents at the vertices A, A' in L and L'. Then AL. A'L' =

  1. a + b    
  2. a2 + b2    
  3. a2    
  4. b2    

hintHint:

assume a point and find out the equation of tangent. find the points of intersection with the tangents  and find the distance of those points from the vertices.

The correct answer is: b2


    b2
    let the point be (t,u)
    equation of tangent : xt/a2+yu/b2=1
    equations of tangents at the vertices : x=a and x=-a
    at x=a, y=(1-t/a)b2/u = L
    at x=-a, y= (1+t/a)b2/u = L’

    AL.AL’ = (1-t2/a2)b4/u2
    put t=a cos Ѳ and u=b sin Ѳ
    this gives us AL.AL’ = b2

    the lines parallel to the y axis are pf the form x=a since the y intercept is undefined.

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