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Q38E

Expert-verifiedFound in: Page 550

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.

**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.

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.

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