# Find the cartesian equation of a curve?

A curve has parametric equations:

x = 2 + t2                           y = 4t

Find the cartesian equation of this curve.

A cartesian equation of a curve is simply finding the single equation of this curve in a standard form where xs and ys are the only variables.

To find this equation, you need to solve the parametric equations simultaneously:

If y = 4t, then divide both sides by 4 to find (1/4)y = t.

This newly found value of t can be substituted into the equation for x:

x = 2 + (1/4(y))2 - expand the bracket (square both 1/4 and y) to derive x = 2 + 1/16 y2

Technically, this final equation is already in cartesian form as it only includes variables x and y, however to further rearrange the equation to find the standard 'y =' form:

x = 2 + 1/16 y2 (minus 2 from both sides)

x - 2 = 1/16 y2 (multiply each side by 16)

16x - 32 = y2    (and finally take square roots of both sides)

y = SQRT(16x-32)

Answered by James F. Maths tutor

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