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Precalculus 7th

David Cohen, Theodore B. Lee, David Sklar

Chapter 12

The Conic Sections

Educators


Problem 1

Exercises $1-12$ are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find the distance between the points $(-5,-6)$ and $(3,-1)$

Raushan K.
Numerade Educator

Problem 2

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find an equation of the line that passes through $(2,-4)$ and is parallel to the line $3 x-y=1 .$ Write your answer in the form $y=m x+b$

Raushan K.
Numerade Educator

Problem 3

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find an equation of a line that is perpendicular to the line $4 x-5 y-20=0$ and has the same $y$ -intercept as the line $x-y+1=0 .$ Write your answer in the form $A x+B y+C=0$

Raushan K.
Numerade Educator

Problem 4

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find an equation of the line passing through the points
$(6,3)$ and $(1,0) .$ Write your answer in the form $y=m x+b$

Raushan K.
Numerade Educator

Problem 5

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find an equation of the line that is the perpendicular bisector of the line segment joining the points $(2,1)$ and $(6,7)$ Write your answer in the form $A x+B y+C=0$

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Problem 6

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find the area of the circle $(x-12)^{2}+(y+\sqrt{5})^{2}=49$

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Problem 7

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find the $x$ - and $y$ -intercepts of the circle with center $(1,0)$ and radius 5

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Problem 8

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find an equation of the line that has a positive slope and is tangent to the circle $(x-1)^{2}+(y-1)^{2}=4$ at one of its $y$ -intercepts. Write your answer in the form $y=m x+b$

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Problem 9

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Suppose that the coordinates of $A, B,$ and $C$ are $A(1,2)$ $B(6,1),$ and $C(7,8) .$ Find an equation of the line passing through $C$ and through the midpoint of the line segment $A B$. Write your answer in the form $a x+b y+c=0$

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Problem 10

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find an equation of the line passing through the point $(-4,0)$ and through the point of intersection of the lines $2 x-y+1=0$ and $3 x+y-16=0 .$ Write your answer in the form $y=m x+b$

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Problem 11

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find the perimeter of $\triangle A B C$ in the following figure.
FIGURE CAN'T COPY.

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Problem 12

Are review exercises. To solve these problems, you will need to utilize the formulas listed at the beginning of this section.
Find the sum of the $x$ - and $y$ -intercepts of the line
$$
\left(\csc ^{2} \alpha\right) x+\left(\sec ^{2} \alpha\right) y=1
$$
(Assume that $\alpha$ is a constant.)

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Problem 13

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
$$y=\sqrt{3} x+4$$

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Problem 14

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
$$x+\sqrt{3} y-2=0$$

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Problem 15

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
(a) $y=5 x+1$
(b) $y=-5 x+1$

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Problem 16

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
(a) $3 x-y-3=0$
(b) $3 x+y-3=0$

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Problem 17

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
$$(1,4) ; y=x-2$$

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Problem 18

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
$$(-2,-3) ; y=-4 x+1$$

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Problem 19

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
$$(-3,5) ; 4 x+5 y+6=0$$

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Problem 20

Determine the angle of inclination of each line. Express the answer in both radians and degrees. In cases in which a calculator is necessary, round the answer to two decimal places.
$$(0,-3) ; 3 x-2 y=1$$

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Problem 21

(a) Find the equation of the circle that has center $(-2,-3)$ and is tangent to the line $2 x+3 y=6$
(b) Find the radius of the circle that has center $(1,3)$ and is tangent to the line $y=\frac{1}{2} x+5$

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Problem 22

Find the area of the triangle with vertices $(3,1),(-2,7)$ and $(6,2) .$ Hint: Use the method shown in Example 3 .

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Problem 23

Find the area of the quadrilateral $A B C D$ with vertices $A(0,0), B(8,2), C(4,7),$ and $D(1,6) .$ Suggestion: Draw a diagonal, and use the method shown in Example 3 for the two resulting triangles.

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Problem 24

From the point $(7,-1),$ tangent lines are drawn to the circle $(x-4)^{2}+(y-3)^{2}=4 .$ Find the slopes of these lines.

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Problem 25

From the point $(0,-5),$ tangent lines are drawn to the circle $(x-3)^{2}+y^{2}=4 .$ Find the slope of each tangent.

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Problem 26

Find the distance between the two parallel lines $y=2 x-1$ and $y=2 x+4 .$ Hint: Draw a sketch; then find the distance from the origin to each line.

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Problem 27

Find the distance between the two parallel lines $3 x+4 y=12$ and $3 x+4 y=24$

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Problem 28

Find an equation of the line that passes through $(3,2)$ and whose $x$ - and $y$ -intercepts are equal. (There are two answers.)

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Problem 29

Find an equation of the line that passes through the point $(2,6)$ in such a way that the segment of the line cut off between the axes is bisected by the point $(2,6)$

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Problem 30

Find an equation of the line whose angle of inclination is $60^{\circ}$ and whose distance from the origin is four units. (There are two answers.)

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Problem 31

Find an equation of the angle bisector in the accompanying figure. Hint: Let $(x, y)$ be a point on the angle bisector. Then $(x, y)$ is equidistant from the two given lines. Or use angles of inclination.

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Problem 32

Find the center and the radius of the circle that passes through the points $(-2,7),(0,1),$ and $(2,-1)$

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Problem 33

(a) Find the center and the radius of the circle passing through the points $A(-12,1), B(2,1)$ and $C(0,7)$
(b) Let $R$ denote the radius of the circle in part (a). In $\triangle A B C,$ let $a, b,$ and $c$ be the lengths of the sides opposite angles $A, B,$ and $C,$ respectively. Show that the area of $\triangle A B C$ is equal to $a b c / 4 R$

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Problem 34

(Continuation of Exercise 33 .)
(a) Let $H$ denote the point where the altitudes of $\triangle A B C$ intersect. Find the coordinates of $H$
(b) Let $d$ denote the distance from $H$ to the center of the circle in Exercise $33($ a ) . Show that
$$d^{2}=9 R^{2}-\left(a^{2}+b^{2}+c^{2}\right)$$

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Problem 35

Suppose the line $x-7 y+44=0$ intersects the circle $x^{2}-4 x+y^{2}-6 y=12$ at points $P$ and $Q .$ Find the length of the chord $\overline{P Q}$.

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Problem 36

The point $(1,-2)$ is the midpoint of a chord of the circle $x^{2}-4 x+y^{2}+2 y=15 .$ Find the length of the chord.

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Problem 37

Show that the product of the distances from the point $(0, c)$ to the lines $a x+y=0$ and $x+b y=0$ is
$$\frac{\left|b c^{2}\right|}{\sqrt{a^{2}+a^{2} b^{2}+b^{2}+1}}$$

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Problem 38

Suppose that the point $\left(x_{0}, y_{0}\right)$ lies on the circle $x^{2}+y^{2}=a^{2}$ Show that the equation of the line tangent to the circle at $\left(x_{0}, y_{0}\right)$ is $x_{0} x+y_{0} y=a^{2}$

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Problem 39

The vertices of $\triangle A B C$ are $A(0,0), B(8,0),$ and $C(8,6)$
(a) Find the equations of the three lines that bisect the angles in $\triangle A B C .$ Hint: Make use of the identity $\tan (\theta / 2)=(\sin \theta) /(1+\cos \theta)$
(b) Find the points where each pair of angle bisectors intersect. What do you observe?

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Problem 40

Show that the equations of the lines with slope $m$ that are tangent to the circle $x^{2}+y^{2}=a^{2}$ are $y=m x+a \sqrt{1+m^{2}} \quad$ and
$y=m x-a \sqrt{1+m^{2}}$

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Problem 41

The point $(x, y)$ is equidistant from the point $(0,1 / 4)$ and the line $y=-1 / 4 .$ Show that $x$ and $y$ satisfy the equation $y=x^{2}$

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Problem 42

The point $\left(x_{0}, y_{0}\right)$ is equidistant from the line $x+2 y=0$ and the point $(3,1) .$ Find (and simplify) an equation relating $x_{0}$ and $y_{0}$

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Problem 43

(a) Find the slope $m$ and the $y$ -intercept $b$ of the line $A x+B y+C=0$
(b) Use the formula $d=\left|m x_{0}+b-y_{0}\right| / \sqrt{1+m^{2}}$ to show that the distance from the point $\left(x_{0}, y_{0}\right)$ to the line $A x+B y+C=0$ is given by$$d=\frac{\left|A x_{0}+B y_{0}+C\right|}{\sqrt{A^{2}+B^{2}}}$$

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Problem 44

Refer to Figure 5 in this section. Show that $\triangle A B D$ is similar to $\triangle E G F$

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Problem 45

Show that the distance of the point $\left(x_{1}, y_{1}\right)$ from the line passing through the two points $\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$ is giver by $d=|D| / \sqrt{\left(x_{2}-x_{3}\right)^{2}+\left(y_{2}-y_{3}\right)^{2}},$ where $$D=\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\x_{2} & y_{2} & 1 \\x_{3} & y_{3} & 1\end{array}\right|$$

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Problem 46

Let $\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right),$ and $\left(a_{3}, b_{3}\right)$ be three noncollinear points. Show that an equation of the circle passing through these three points is$$\left|\begin{array}{llll}
x^{2}+y^{2} & x & y & 1 \\a_{1}^{2}+b_{1}^{2} & a_{1} & b_{1} & 1 \\a_{2}^{2}+b_{2}^{2} & a_{2} & b_{2} & 1 \\a_{3}^{2}+b_{3}^{2} & a_{3}&b_{3}& 1\end{array}\right|=0$$

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Problem 47

Find an equation of the circle that passes through the points $(6,3)$ and $(-4,-3)$ and that has its center on the line $y=2 x-7$

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Problem 48

Let $a$ be a positive number and suppose that the coordinates of points $P$ and $Q$ are $P(a \cos \theta, a \sin \theta)$ and $Q(a \cos \beta, a \sin \beta) .$ Show that the distance from the origin to the line passing through $P$ and $Q$ is
$$a\left|\cos \left(\frac{\theta-\beta}{2}\right)\right|$$

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Problem 49

Find an equation of a circle that has radius 5 and is tangent to the line $2 x+3 y=26$ at the point $(4,6) .$ Write your answer in standard form. (There are two answers.)

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Problem 50

Find an equation of the circle passing through $(2,-1)$ and tangent to the line $y=2 x+1$ at $(1,3) .$ Write your answer in standard form.

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Problem 51

For the last exercise in this section you will prove an interesting property of the curve $x^{3}+y^{3}=6 x y .$ This curve, known as the folium of Descartes, is shown in Figure A.
FIGURE CAN'T COPY.
A property of the folium: Suppose that a line through the point $P(3,3)$ meets the folium again at points $Q$ and $R$. Then $\angle R O Q$ is
a right angle.
Figure $\mathbf{B}$
Figure $\mathrm{B}$ and the accompanying caption indicate the property that we will establish.
(a) Suppose that the slope of the line segment $\overline{O Q}$ is $t$ By solving the system of equations $$\left\{\begin{array}{l}y=t x \\
x^{3}+y^{3}=6 x y\end{array}\right.$$show that the coordinates of the point $Q$ are $x=6 t /\left(1+t^{3}\right)$ and $y=6t^{2}/\left(1+t^{3}\right)$
(b) Show that the slope of the line joining the points $P$ and $Q$ is $\left(t^{2}-t-1\right) /\left(t^{2}+t-1\right)$
(c) Suppose that the slope of the line segment $\overline{O R}$ is $u$ By repeating the procedure used in parts (a) and (b), you'll find that the slope of the line joining the points $P$ and $R$ is $\left(u^{2}-u-1\right) /\left(u^{2}+u-1\right)$. Now, since the points $P, Q,$ and $R$ are collinear (lie on the same line), it must be the case that
$$\frac{t^{2}-t-1}{t^{2}+t-1}=\frac{u^{2}-u-1}{u^{2}+u-1}$$
Working from this equation, show that $t u=-1 .$ This shows that $\angle R O Q$ is a right angle, as required.

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