# Suppose a curve is given by the parametric equations x = f ( t ) , y = g ( t ) . where the...

## Question:

Suppose a curve is given by the parametric equations {eq}x = f(t), \ y = g(t) {/eq}. where the range of {eq}f {/eq} is {eq}[1,7] {/eq} and the range of {eq}g {/eq} is {eq}[-2,7] {/eq}. What can you say about the curve? You may select multiple answers.

A. The curve is a circle with center {eq}(1,-2) {/eq} and radius {eq}7 {/eq}.

B. The curve is the line with endpoints {eq}(1, -2) {/eq} and {eq}(7,7) {/eq}.

C. The curve must lie inside a circle with center {eq}(1,-2) {/eq} and radius {eq}0.5 {/eq}.

D. Nothing can be said about the curve.

E. The curve is completely contained in the rectangle {eq}[1,7] {/eq} by {eq}[-2,7] {/eq}.

F. The curve must lie outside the rectangle {eq}[1,7] {/eq} by {eq}[-2,7] {/eq}.

## Parametric Curves:

{eq}\\ {/eq}

If we have a parametric curve given by : {eq}x=f(t), y=g(t) {/eq} and the range of {eq}f {/eq} is {eq}[a,b] {/eq} and range of {eq}g {/eq} is {eq}[c,b] {/eq}, the region is enclosed in the rectangle with vertices {eq}(a,c),(a,b),(b,c) \ \&(b,b). {/eq}

{eq}\\ {/eq}

From the given information it is clear that the curve lies inside the rectangle with vertices {eq}(1,-2),(1,7),(7,-2) \ \& (7,7) {/eq} i.e. curve completely lies within the rectangle {eq}[1,\ 7] {/eq} by {eq}[-2,\ 7] {/eq}.

So, option {eq}(E) {/eq} is correct. 