  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) (13,1√3) (14,12) (13,−1√3) (14,−12)

Solution

## The correct options are A (13,1√3) C (13,−1√3) 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=1−cosθsinθ∴ Q(1,1−cosθsinθ) ∴  Equation of the line through Q parallel to RS is y=1−cosθsinθ=2sin2θ22sinθ2cosθ2=tanθ2……(i) Normal at P:  y=sinθcosθ.x⇒y=xtanθ……(ii) Let their point of intersection be (h, k) Then k=tanθ2 and k=h tanθ∴ k=h(2tanθ21−tan2θ2)⇒k=2h.k1−k2⇒ k(1−k2)=2hk ∴  Locus for point E: 2x=1−y2……(iii) When x=13, then 1−y2=23⇒y2=1−23⇒ y=±1√3 ∴ (13,±1√3)  satisfy 2x=1−y2 When x=14, then 1−y2=24⇒ y2=1−12⇒ y=±1√2 ∴ (14,±12) does not satisfy 1−y2=2x  Suggest corrections   