How to find the cartesian equation from parametric equation?

enter image source here
Can someone please explain to me how to do question 2? Thanks!

2 Answers
Dec 15, 2017

x=y^2/16

Explanation:

We know that x=4t^2 and y=8t. We're going to eliminate the parameter t from the equations.

Since y=8t we know that t=y/8.

We can now substitute for t in x=4t^2:

x=4(y/8)^2\rightarrow x=(4y^2)/64\rightarrow x=y^2/16

Although it is not a function, x=y^2/16 is a form of the Cartesian equation of the curve. It's frequently the case that you do not end up with y as a function of x when eliminating the parameter from a set of parametric equations.

Dec 15, 2017

"see explanation"

Explanation:

(a)

y=8trArrt=1/8y

rArrx=4t^2=4xx(1/8y)^2=1/16y^2

rArrx=1/16y^2larrcolor(blue)"cartesian equation"

(b)color(white)(x)"substitute values of t into x and y"

t=1tox=4,y=8rArr(4,8)

t=-1tox=4,y=-8rArr(4,-8)

"the equation of the line passing through"

(color(red)(4),8)" and "(color(red)(4),-8)" is "x=4

(c)color(white)(x)" substitute values of t into x and y"

t=1tox=4,y=8rArr(4,8)

t=-3tox=36,y=-24rArr(36,-24)

"calculate the length using the "color(blue)"distance formula"

•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

rArrd=sqrt((36-4)^2+(-24-8)^2)

color(white)(rArrd)=sqrt2048=32sqrt2