Question

1.
Find the locus of the point of intersection of tangents to the circle x = acos 0, y = asin at the
points whose parametric angles differ by (i) 7/3, (ii) 7/2.
&ty ²​

Answer 1

Answer:

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Answer 2

Answer:

Let one of the points on the circle be A(acosθ,asinθ)

Then the other point will be B(acos(θ+π),asin(θ+π))

Therefore equation of tangent at A=xcosθ+ysinθ=a...(1)

Equation of tangent at B=xcos(θ+π)+ysin(θ+π)=a...(2)

From (2)

⇒x[

2

1

cosθ−

2

3

sinθ]+y[

2

1

sinθ+

2

3

cosθ]=a

A(acosθ,asinθ)

B(acos(θ+π),asin(θ+π))

A=xcosθ+ysinθ=a...(1)

⇒x[

2

1

cosθ−

2

3

sinθ]+y[

2

1

sinθ+

2

3

cosθ]=a

⇒

2

1

(xcosθ+ysinθ)−

2

3

(xsinθ−ycosθ)=a

⇒(xsinθ−ycosθ)=

3

−a

...(3)

Square and add (1) and (3)

(xcosθ+ysinθ)

2

+(xsinθ−ycosθ)

2

=a

2

+

3

a

2

⇒3x

2

+3y

2

=4a

2

The locus of the point of intersection of the tangents 3x

2

+3y

2

−4a

2

=0