Parametric to implicit form of a curve

Question

"Find the implicit form of the curve defined by parametric equations $x = t+1,y=\frac{1}{t^{2}}$"

How can I clear $t$ to arrive at the implicit equation?

Answer 1

HINT:

If $x=\dfrac1{t+1}\iff t+1=\dfrac1x\iff t=\cdots$

If $x=t+1\iff t=x-1$

Put this value in $y=\dfrac1{t^2}$

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Alternatively, $y=\dfrac1{t^2}\iff t^2=\dfrac1y\ \ \ \ (1)$

and $x=t+1\iff t=x-1\implies t^2=(x-1)^2\ \ \ \ (2)$

Compare the values of $t^2$ in $(1),(2)$ to eliminate $t$

Answer 2

Another approach:

$$y=\frac1{t^2}=\frac1{(t+1)^2-(2t+1)}=\frac1{x^2-(2x-1)}=\frac1{(x-1)^2}$$

and we're done.

Answer 3

Starting from $$ x=t+1 \quad y=\frac{1}{t^2} $$ we get $$ y = \frac{1}{(x-1)^2} $$ and thus $$ F(x,y) = \frac{1}{(x-1)^2} - y = 0 $$ as implicit curve equation.

Answer 4

An approach: express $t$ as a function of the other variables.

$$t=x-1=\pm\frac1{\sqrt y}.$$

In this case, it is better to express $t^2$ to avoid the double signs:

$$t^2=(x-1)^2=\frac1y.$$

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Alternatively, say: "take $t+1$, subtract $1$ to get $t$, square and take the inverse to get $\dfrac1{t^2}$."

Then $\dfrac1{(x-1)^2}=y$.