Question

The locus of the point of intersection of the tangents to the circle `x^2+ y^2 = a^2` at points whose parametric angles differ by `pi/3`.

Answer 1

Let the points on the circle whose parametrci angles differ by `60^(@)` be P (a cos `theta`,a sin `theta`) and `Q(a cos (theta+60^(@))`, `a sin ( theta+60^(@))`. Tangents at points P and Q intersect at R(h,k).
In the figure, `/_POQ=60^(@)` and `/_POR=30^(@)`.
In triangle OPR,
`OP=OR cos 30^(@)`
`:. a=sqrt(h^(2)+k^(2))(sqrt(3))/(2)`
Hence, locus of R is `3(x^(2)+y^(2))=4`