Home
Class 11
MATHS
The locus of the point of intersection o...

The locus of the point of intersection of the tangents to the circle `x=a cos theta, y =a sin theta` at the points, whose parametric angles differ by `(pi)/(3)` is

A

Straight line

B

Ellipse

C

Circle of radius 2a

D

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

Text Solution

Verified by Experts

The correct Answer is:
D
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)`
Promotional Banner

Similar Questions

Explore conceptually related problems

Find the locus of the point of intersection of tangents to the circle x=a cos theta,y=a sin theta at the points whose parametric angles differ by (i)(pi)/(3)

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) .

The locus of points of intersection of tangents to the circle x^(2)+y^(2)=a^(2) at the point whose parametric angle differ why (4 pi)/(3) is a director circle of circle whose radius is

Locus of the point of intersection of the tangents to the circle x^(2)+y^(2)=a^(2), at the points whose parameters differ by alpha, is

The locus of the point of intersection of tangents to the ellipse x^(2)/a^(2) + y^(2)/b^(2) = 1 at the points whose eccentric angles differ by pi//2 , is

The equation of the tangent to the curve x=2cos^(3) theta and y=3sin^(3) theta at the point, theta =pi//4 is

Find the equations of tangent to the curve x=1-cos theta,y=theta-sin theta at theta=(pi)/(4)

The equation of tangent to the curve x=a cos^(3) theta, y= a sin^(3) theta" at "theta=(pi)/(4) is