Whatever be the value of theta, prove that the locus of the point of intersection of the straight lines #y=xtan(theta)# and #xsin^3(theta)+ycos^3(theta)=asin(theta)cos(theta)# is a circle.find the equation of the circle?

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Answer 1 · 2018-02-15T12:03:50

See below.

Solving for #x,y#

#{(y=x tan theta),(x sin^3 theta+y cos^3 theta = a sin theta cos theta):}#

we get

#x = a cos theta# and #y = a sin theta#

which is the polar form for the circle

#x^2+y^2=a^2#

Answer 2 · 2018-03-21T12:20:15

Given

#{(y=x tan theta......[1]),(x sin^3 theta+y cos^3 theta = a sin theta cos theta.....[2]):}#

Dividing both sides of [2] by #cos^3theta# we get

#x sin^3 theta/cos^3theta+y cos^3 theta /cos^3theta= a (sin theta cos theta)/cos^3theta#

#=>x tan^3 theta+y = a tan theta sec theta#

#=>x tan^3 theta+y = a tan thetasqrt (1+tan^2theta) #

#=>x y^3/x^3+y = ay/x sqrt(1+y^2/x^2) #

#=>y(y^2/x^2+1)= ay/x sqrt(1+y^2/x^2) #

#=>y(sqrt(y^2/x^2+1))^2= ay/x sqrt(1+y^2/x^2) #

#=>sqrt(y^2/x^2+1)= a/x #

#=>y^2/x^2+1= a^2/x^2 #

#=>x^2+y^2= a^2 #

This is the equation of the circle