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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

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

Grade:11

1 Answers

Vikas TU
14149 Points
7 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

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