Analyse the following equations if they represent parabola(s) or part of parabola(s) ?
(a) `y=x|x|` , (b) `x=sqrt(-y)`
(c ) `x^(2)=y^(4)` , (d) `x=e^(t),2t=log_(e)y`

Explanation

(a) `y=x|x|`
`:." "y={{:(x*(-x)",",xlt0),(x*x",",xge0):}`
`or" "y={{:(-x^(2)",",xlt0),(x^(2)",",xge0):}`
So, equation represents parts of two parabolas as shown in the figure.

(b) `x=sqrt(-y),yle0`
`:." "x^(2)=-y,yle0`
So, equation represents part of the parabola lying in fourth quadrant.

(c ) `x^(2)=y^(4)`
`:." "y^(2)=pmx`
So, equation represents two parabolas.

(d) `x=e^(t),2t=log_(e)y`
`:." "x=e^(t),y=e^(2t)`
`:." "y=x^(2),xgt0`
So, equation represents part of the parabola lying in first quadrant.