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

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

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

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

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

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

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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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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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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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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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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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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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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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