Modern Analytic Geometry

William Wooton, Edwin F. Beckenbach, Frank J. Fleming

Chapter 4

Conic Sections - all with Video Answers

Educators

Section 1

Equation of a Locus

Problem 1

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of $\mathcal{C}$ is equidistant from the points $\mathbf{S}(1,4)$ and $\mathbf{T}(3,7)$.

Debasish Das

Debasish Das

Numerade Educator

Problem 2

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of $\mathcal{C}$ is equidistant from the points $\mathbf{S}(-3,2)$ and $\mathbf{T}(2,-5)$.

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Problem 3

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of $\mathcal{C}$ is 6 units from the point $\mathbf{S}(2,4)$.

Debasish Das

Numerade Educator

Problem 4

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of e is $\sqrt{7}$ units from the point $S(-1,-3)$.

Amrita Bhasin

Numerade Educator

Problem 5

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The line segment connecting each point of $\mathcal{e}$ with $S(-3,0)$ is perpendicular to the line segment connecting the point with $\mathbf{T}(3,0)$.

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Problem 6

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Repeat Exercise 5 for $\mathbf{S}(1,2)$ and $\mathbf{T}(5,-2)$.

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Problem 7

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The slope of the line segment connecting each point of e with $\mathbf{S}(1,6)$ is twice the slope of the line segment connecting the point with $\mathbf{T}(3,2)$.

Jay Patel

Numerade Educator

Problem 8

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The slope of the line segment connecting each point of $\mathcal{C}$ with $\mathbf{S}(2,-3)$ is two-thirds of the slope of the line segment connecting the point with $\mathbf{T}(-1,4)$

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Problem 9

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The slope of the line segment connecting each point of $\mathcal{C}$ with $\mathbf{S}(2,5)$ is 2 more than the slope of the line segment connecting the point with $\mathbf{T}(-1,2)$.

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Problem 10

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The slope of the line segment connecting each point of $\mathcal{C}$ with $\mathbf{S}(1,-2)$ is 3 less than the slope of the line segment connecting the point with $\mathbf{T}(-1,-2)$.

WM

William Mead

Numerade Educator

Problem 11

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of $\mathcal{e}$ is equidistant from $\mathbf{S}(6,0)$ and the $y$-axis.

Jay Patel

Numerade Educator

Problem 12

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of $\mathfrak{e}$ is equidistant from $\mathbf{S}(0,2)$ and the line with equation $y+2=0$.

Jay Patel

Numerade Educator

Problem 13

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of e is equidistant from $\mathbf{S}(1,2)$ and the line with equation $x-y-5=0$

Jay Patel

Numerade Educator

Problem 14

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
Each point of $\mathcal{C}$ is equidistant from $S(3,-2)$ and the line with equation $x-y=0$

Jay Patel

Numerade Educator

Problem 15

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The distance from each point of $\mathcal{C}$ to $S(4,0)$ is one-half the distance from the point to the line with equation $x+8=0$.

Debasish Das

Numerade Educator

Problem 16

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The distance from each point of $\mathcal{C}$ to $S(4,0)$ is twice its distance to the line with equation $x+8=0$

Debasish Das

Numerade Educator

Problem 17

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The sum of the distances from each point of $\mathcal{e}$ to $S(-3,0)$ and $T(3,0)$ is 10 .

Debasish Das

Numerade Educator

Problem 18

find an equation for the locus $\mathfrak{C}$ of all points in the plane satisfying the stated conditions.
The absolute value of the difference of the distances from each point of $\mathcal{C}$ to $\mathbf{S}(-5,0)$ and $\mathbf{T}(5,0)$ is 8 .

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Problem 19

Let $\mathbf{U}$ be any point on the graph of $y=x^2$, and let $\mathbf{S}$ be the (perpendicular) projection of $\mathbf{U}$ on the $x$-axis. Find an equation for the locus of the midpoint $\mathbf{M}$ of $\overline{\mathbf{U S}}$.
Figure Can't Copy

Carson Merrill

Numerade Educator

Problem 20

Let $\mathbf{U}$ be any point on the graph of $y=x^2$, and let $\mathbf{S}$ and $\mathbf{T}$ be the (perpendicular) projections of $\mathbf{U}$ on the $x$-and $y$-axes, respectively. Find an equation for the locus of the midpoint $\mathbf{M}$ of $\overline{\mathbf{S T}}$.

Carson Merrill

Numerade Educator

Problem 21

Let $\mathbf{U}$ be any point on the graph of $y=3 x^2$, and let $\mathbf{S}$ and $\mathbf{T}$ be the (perpendicular) projections of $\mathbf{U}$ on the $x$ - and $y$-axes, respectively. Find an equation for the locus of the point $\mathbf{Q}$ on $\overline{\mathbf{S T}}$ that lies one-third of the way from $\mathbf{S}$ to $\mathbf{T}$. [Hint: Draw a sketch similar to the one for Exercise 20.]

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