Q2. Whatever be the values of θ, prove that the locus of the point of intersection of the straight lines y =atanθ and asin3θ + ycos3θ = asinθcosθ is a circle. Find the equation of the circle.

put x=3, y=1 in the eq. U get 9+1-3(3p+4)-(p-2)+10p=12-12-10p+10p=0.

ie. For any value of p the pt.(3,1) satisfy the eq. Hence proved.

Let center be (h,k). From eq.

h=(3p+4)/2 => p=(2h-4)/3

k=(p-2)/2 . Sub. value of p we get:

3k-h+5=0

ie. locus of center is:

3y-x+5=0 ie. straight line