can somebody explain me how to solve this problem about parametric equations?

Question

all of the following sets of parametric equations will generate a different curve. However, four of the five will have the same graph. Identify the set of parametric equations that will produce a graph diffferent from the others. show your work to justify your answer.

a) x(t)=2cos(t) y(t)=7sint(t) 0 less or equal t less or equal 2pi

b)x(t)2sin(t) y(t)=7cos(t) 0 less or equal t less or equal 2pi

c)x(t)2cos(2t) y(t)=7sin(2t) 0 less or equal t less or equal pi

d)x(t)2cos(t/3) y(t)=7sin(t/3) 0 less or equal t less or equal 3pi

e)x(t)2cos(t) y(t)= -7sin(t) 0 less or equal t less or equal 2pi

Answer 1

I will show you how to do the first one and all the rest are done the same way.

Make cos and sin the subject of each equation and use the identity cos²A + sin²A = 1

for(a) cost = x/2 sint = y/7

so cos²t + sin²t = 1 means that (x/2)² + (y/7)² = 1

You will get the same answer for all the others. The difference will lie in the limits of the value of t.

a), b) and e) will all be the same complete loop of the ellipse.
c) does the complete loop as well because when t = π you will have arrived at 2cos 2π and 7sin2π.
d) is the odd one out because it only reaches half way round the ellipse to 2cosπ and 7sinπ

Answer 2

d. Note that all the other equations generate all points satisfying the relation (x/2)² + (y/7)² = 1 -- that is, the ellipse with vertical semi-major axis 7 and semi-minor axis 2. However, d only generates half those points (specifically, the half lying above the x-axis). Therefore, d is the one that generates a different graph from the others.

Answer 3

sparkling up between the equations for t (the better finding one) and substitute that answer for each t in the different equation. After simplifying, this could provide you a classic equation to sparkling up