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Question

Let RS be the diameter of  the circle x2+y2=1, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E.  Then, the locus of E passes through the point(s)


  1. (13,13)

  2. (14,12)

  3. (13,13)

  4. (14,12)


Solution

The correct options are
A

(13,13)


C

(13,13)


Given, RS is the diameter of x2+y2=1
Here, equation of the tangent at p(cosθ,sinθ) is xcos θ+ysin θ=1


This tangent intersects with the tangent x=1
y=1cosθsinθ Q(1,1cosθsinθ)
  Equation of the line through Q parallel to RS is
y=1cosθsinθ=2sin2θ22sinθ2cosθ2=tanθ2(i)
Normal at P:  y=sinθcosθ.xy=xtanθ(ii)
Let their point of intersection be (h, k)
Then k=tanθ2 and k=h tanθ k=h(2tanθ21tan2θ2)k=2h.k1k2 k(1k2)=2hk
  Locus for point E: 2x=1y2(iii)
When x=13, then
1y2=23y2=123 y=±13
 (13,±13)  satisfy 2x=1y2
When x=14, then
1y2=24 y2=112 y=±12
 (14,±12) does not satisfy 1y2=2x

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