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Q38E

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Found in: Page 550

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# To describe the set of all points for condition $$\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k$$.

All set of points for condition $$\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k$$ are described and it represents an ellipse.

See the step by step solution

## Step 1: Concept of Two-Dimensional Vector

The magnitude of a vector refers to the entire amount of the quantity it represents. The magnitude of a two-dimensional vector is equal to the length of the hypotenuse of a triangle with the $${\bf{x}}$$ - and $$\;{\bf{y}}$$ -components as sides.

Given:

Three two-dimensional vectors $${\bf{r}} = \langle x,y\rangle ,{{\bf{r}}_1} = \left\langle {{x_1},{y_1}} \right\rangle$$ and $${{\bf{r}}_2} = \left\langle {{x_2},{y_2}} \right\rangle$$.

Formula used:

Write the expression for ellipse with center $$C(h,k)$$.

$$\frac{{{{(x - h)}^2}}}{{{a^2}}} + \frac{{{{(y - k)}^2}}}{{{b^2}}} = 1.......(1)$$

Here,

$$x,{\rm{ }}y$$, and $$z$$ are Cartesian coordinates.

## Step 2: Calculate sets of all points for condition

Write the expression for relation between vectors $${\bf{r}},{{\bf{r}}_1}$$ and $${{\bf{r}}_2}$$.

$$\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k.......(2)$$

Consider $${{\bf{r}}_1}$$ and $${{\bf{r}}_2}$$ as points $${P_1}$$ and $${P_2}$$ respectively. The equation (2) indicates the sum of distances between $$(x,y)$$ and points $${P_1}$$ and $${P_2}$$. According to the equation (1) set of points in equation (2) represents foci of ellipse. The no degenerating condition of ellipse is $$k > \left| {{{\bf{r}}_1} - {{\bf{r}}_2}} \right|$$.

Thus, all set of points for condition $$\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k$$ are described and it represents an ellipse.