Question

eliminate the parabola and hence find the cartesian equation of the curve with x=1+2tantheta and y=3sectheta-4

Answer 1

"...and hence..." I seem to remember that redundant statement being in JonBenet Ramsey's hostage letter. This might help: http://community.boredofstudies.org/12/mathematics/289228/locus-parabola-help.html

Answer 2

x=1+2tantheta
tantheta =(x-1)/2
sec^2theta=1+tan^2(theta)
=1+((x-1)/2)^2
y=3sectheta-4
3sectheta=y+4
square on both sides
9sec^2theta=(y+4)^2
9(1+((x-1)/2)^2)=(y+4)^2
9=(y+4)^2-((x-1)^2/4
(y+4)^2/9-((x-1)^2/36 =1
Source: http://www.mathskey.com/question2answer/