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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) (13, √3), (14, √2), (14, 1/2), (13, 1/√3)

(13,1√3)

(13,√3)

(14,√2)

(14,12)

Solution:

We can see the line on which point E lies is having equation y = tanθx. Now as we can see they coordinate of E is (1−cosθsinθ). Substituting in the equation we get the coordinate of E as ((1−cosθsinθ)tanθ), (1−cosθsinθ). E((1−cosθsinθ)tanθ), (1−cosθsinθ)) ⇒E(tanθ2tanθ,tanθ2) Let h = tanθ2tanθ and k = tanθ2 ∴h=ktanθ ∴tanθ2 = k/n