Let 𝐫1=⟨x1, y1⟩, 𝐫2=⟨x2, y2⟩, and 𝐫=⟨x, y⟩. As- suming that k>𝐫2-𝐫1, describe the set of all poi

Name: Problem 44, Chapter 11: Calculus Early Transcendentals
Uploaded: 2026-07-03T22:42:19-08:00
Duration: 1 min
Channel: Nick Johnson
Description: Let 𝐫1=⟨x1, y1⟩, 𝐫2=⟨x2, y2⟩, and 𝐫=⟨x, y⟩. As- suming that k>𝐫2-𝐫1, describe the set of all points (x, y) for which 𝐫-𝐫1+𝐫-𝐫2=k

step 1 Hello. So this is giving the definition of an ellipse where we have the condition, the magnitude

Question

Let $\mathbf{r}_{1}=\left\langle x_{1}, y_{1}\right\rangle, \mathbf{r}_{2}=\left\langle x_{2}, y_{2}\right\rangle$, and $\mathbf{r}=\langle x, y\rangle .$ As- suming that $k>\left\|\mathbf{r}_{2}-\mathbf{r}_{1}\right\|$, describe the set of all points $(x, y)$ for which $\left\|\mathbf{r}-\mathbf{r}_{1}\right\|+\left\|\mathbf{r}-\mathbf{r}_{2}\right\|=k$

   Let $\mathbf{r}_{1}=\left\langle x_{1}, y_{1}\right\rangle, \mathbf{r}_{2}=\left\langle x_{2}, y_{2}\right\rangle$, and $\mathbf{r}=\langle x, y\rangle .$ As-
suming that $k>\left\|\mathbf{r}_{2}-\mathbf{r}_{1}\right\|$, describe the set of all points $(x, y)$ for which $\left\|\mathbf{r}-\mathbf{r}_{1}\right\|+\left\|\mathbf{r}-\mathbf{r}_{2}\right\|=k$

Calculus Early Transcendentals

Howard Anton, Irl C.… 10th Edition

Chapter 11, Problem 44

Instant Answer

Solved by Expert Nick Johnson

07/03/2026

Step 1

If $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$, then the distance between $\mathbf{a}$ and $\mathbf{b}$ is given by $\|\mathbf{a} - \mathbf{b}\| = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}$. Now, let's consider the given Show more…

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Let $\mathbf{r}_{1}=\left\langle x_{1}, y_{1}\right\rangle, \mathbf{r}_{2}=\left\langle x_{2}, y_{2}\right\rangle$, and $\mathbf{r}=\langle x, y\rangle .$ As- suming that $k>\left\|\mathbf{r}_{2}-\mathbf{r}_{1}\right\|$, describe the set of all points $(x, y)$ for which $\left\|\mathbf{r}-\mathbf{r}_{1}\right\|+\left\|\mathbf{r}-\mathbf{r}_{2}\right\|=k$

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Key Concepts

Euclidean Distance

Euclidean distance is the measure of the straight-line distance between two points in a plane, calculated using the distance formula derived from the Pythagorean theorem. It is essential in defining the geometrical properties of shapes such as ellipses, where distances from points to fixed locations (the foci) determine the locus of the shape.

Ellipse

An ellipse is a set of points in a plane for which the sum of the distances from any point on the curve to two fixed points (known as the foci) is constant. This geometric definition is fundamental in analytic geometry and requires that the constant is greater than the distance between the two foci to form a proper ellipse.

Foci

Foci are the two fixed points used in the definition of an ellipse. For any point on the ellipse, the sum of its distances to these foci is constant. The properties of the ellipse are deeply tied to the positions of these foci.

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