Paragraph: Let S be the circle in the x y – plane defined by the equation x 2   +   y 2…

Paragraph: Let S be the circle in the xy – plane defined by the equation x2 + y2 = 4.
Question : Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve
  1. x+y2=3xy
  2. x23+y23=243
  3. x2+y2=2xy
  4. x2+y2=x2y2

Solution


Tangent at P2cos⁡θ,2sin⁡θis xcos⁡θ+ysin⁡θ=2
M2sec⁡θ,0 and N0,2cosce⁡θ
Let midpoint be (h, k)
h=sec⁡θ,k=cosce⁡θ
1h2+1k2=1
1x2+1y2=1

Asked in: JEE Advanced 2018 (Paper 1)

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