Which one of the following equation represent parametric equation to a parabolic curve? (a) `x=3cost ; y=4sint` (b) `x^2-2=2cost ; y=4cos^2t/2` (c)`sqrt(x)=tant ;sqrt(y)=sect` (d) `x=sqrt(1-sint ;)y=sint/2+cost/2`

Options

A. `x=3cost,y=4sint`
B. `x^(2)-2=2cost,y=4"cos"^(2)(t)/(2)`
C. `sqrt(x)=tant,sqrt(y)=sect`
D. `x=sqrt(1-sint),y="sin"(t)/(2)+"cos"(t)/(2)`

Answer

B

Explanation

(2) `x=3cost,y=4sint`
Eliminating t, we have
`(x^(2))/(9)+(y^(2))/(16)=1`
which is an ellipse.
`x^(2)-2=2cost andy=4"cos"^(2)(t)/(2)`
`or" "y=2(1+cost)`
`and" "y=2(1+(x^(2)-2)/(2))`
which is a parabola.
`sqrt(x)=tant,sqrt(y)=sect`
Eliminating t, we have
y-x=1
which is a straight line.
`x=sqrt(1-sint)`
`y="sin"(t)/(2)+"cos"(t)/(2)`
Eliminating t, we have `x^(2)+y^(2)=1-sint+1+sint=2`
which is a circle.