Grade 11Analytical Geometry

find the locus of the point of intersection of tangents to the circle x=acosd, y=asindat the points whose parametric angles differ by 60degreee and 90 degree

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10 Years agoGrade 11

1 Answer

Vikas TU

10 Years ago

The parametric equations define the circle of radius a, centered on the origin. When you refer to parametric angles of the tangents, I will suppose you mean the parameters corresponding to their points of tangency.

Let P and Q be the moving points of tangency, where angle POQ = 60°.

Let R be the intersection of the tangents. You are looking for the locus of R.

angle OPR = angle OQR = 90°

That makes OPRQ a cyclic quadrilateral, so angle PRQ = 120°, as opposite angles are supplementary.

By symmetry OR must bisect angles POQ and PRQ.

Therefore, angle ROP = 30°, and ∆RPO is a right triangle.

OR = 2OP = 2a/√(3)

The locus of R is the circle with radius 2a/√(3), centered on the origin