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Find the locus of the point of intersect...

Find the locus of the point of intersection of tangents to the circle `x=acostheta,y=asintheta` at the points whose parametric angles differ by `(i) pi/3, `

A

Straight line

B

Ellipse

C

Circle of radius 2a

D

Circle of radius `(2a)/(sqrt(3))`

Text Solution

Verified by Experts

Equation of tangent at `A(a cos theta, a sin theta)` is `x cos theta + y sin theta=a`….(i)
Equation of tangent at `B[a cos (theta + (pi)/(3)), a sin (theta + (pi)/(3))]`
`x cos (theta + (pi)/(3)) +y sin (theta + (pi)/(3))=a`
`x[(1)/(2) cos theta -(sqrt3)/(2) sin theta] +y[(1)/(2) sin theta + (sqrt3)/(2) cos theta]=a`
`x cos theta +y sin theta + sqrt3(-x sin theta + y cos theta) = 2a`
`a+ sqrt3 (-x sin theta +ycos theta) 2a`
`-x sin theta +y cos theta= (a)/(sqrt3)`....(ii)
Squaring equation (i) and (ii) and adding
`x^(2) +y^(2)=a^(2) + (a^(2))/(3)=(4a^(2))/(3)`
Locus is a circle of radius `(2a)/(sqrt3)`
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