Graphs

For the line $2x+3y=6$, find a line parallel to it containing the point $\left(1,-1\right)$.Also find a line perpendicular to it containing the point $\left(0,3\right)$.

Find an equation of the line having the given characteristics. Express your answer using either the general form or the

slope–intercept form of the equation of a line, whichever you prefer. Graph the line.

y intercept is $-2$ containing the point $\left(5,-3\right)$.

Find an equation of the line having the given characteristics. Express your answer using either the general form or the
slope–intercept form of the equation of a line, whichever you prefer. Graph the line.

$y-$intercept = $-2$; containing the point $\left(5,-3\right)$

Find an equation of the line having the given characteristics. Express your answer using either the general form or the

slope–intercept form of the equation of a line, whichever you prefer. Graph the line.

Containing the points $\left(3,-4\right)$ and $\left(2,1\right)$.

Find an equation of the line having the given characteristics. Express your answer using either the general form or the

slope–intercept form of the equation of a line, whichever you prefer. Graph the line.

Parallel to the line $2x-3y=-4$containing the point $\left(-5,3\right)$.

Find an equation of the line having the given characteristics. Express your answer using either the general form or the

slope–intercept form of the equation of a line, whichever you prefer. Graph the line.

Perpendicular to the line $3x-y=-4$ containing the point $\left(-2,4\right)$ .

 Find the slope and y-intercept of line $4x+6y=36$.

Find the slope and y-intercept of  line.

4x+6y=36

$4x+6y=36$

Find the slope and $y$ intercept of the given line $\frac{1}{2}x+\frac{5}{2}y=10$.

Find the slope and y-intercept of  line.

$\frac{1}{2}x+\frac{5}{2}y=10$

Find the standard form of the equation of the circle whose center and radius are given.

$\left(h,k\right)=\left(-2,3\right)$ and $r=4$

Find the standard form of the equation of the circle whose center and radius are given.

$\left(h,k\right)=\left(-2,3\right)$and $r=4$$r=4$

 Find the standard form of the equation of the circle whose center and radius are given.

$\left(h,k\right)=\left(-1,-2\right)$ and $r=1$

Find the standard form of the equation of the circle whose center and radius are given.

$\left(h,k\right)=\left(-1,-2\right)$and $r=1$

Find the center and radius of each circle. Graph each circle by hand. Determine the intercepts of the graph of circle

$x^2+y^2-2x+4y-4=0$

Find the center and radius of each circle. Graph each circle by hand. Determine the intercepts of the graph of circle .

$3x^2+3y^2-6x+12y=0$

Show that the points $A=\left(-2,0\right) , B=\left(-4,4\right)$ and $C=\left(8,5\right)$

 are the vertices of a right triangle in two ways:

(a) By using the converse of the Pythagorean Theorem .

(b) By using the slopes of the lines joining the vertices .

Show that the points $A\left(2,5\right) , B\left(6,1\right)$ and

$C\left(8,-1\right)$ lie on a straight line by using slopes.

Show that the points $A\left(1,5\right) ,B\left(-3,5\right)$ and

$C\left(-3,5\right)$ lie on a circle with center $\left(-1,2\right)$. What is the radius of the circle.

The endpoints of the diameter of a circle are  $\left(-3,2\right)$and $\left(5,-6\right)$. Find the center and radius of the circle .Write the general equation of this circle.

Find two numbers $y$ such that the distance from$\left(-3,2\right)$ to $\left(5,y\right)$ is $10$.

Graph the line with slope $\frac{2}{3}$and containing the point $\left(1,2\right)$.

Make up four problems that you might be asked to do given the two points $\left(-3,4\right)$ and $\left(6,1\right)$. Each problem should involve a   different concept.Be sure that the directions are clearly stated.

(a) Find the equation of line .Graph the line 

(b) Find the slope for the equation of line .

(c) Find the center between two points.

(d) Find the distance between given points.

Describe each of the following graphs in the $xy$plane. Give justification.

(a) $x=0$     (b) $y=0$

(c) $x+y=0$ (d) $xy=0$

(e) $x^2+y^2=0$

Find the area of the square in the figure.

Find the area of the blue shaded region in the figure, assuming

the quadrilateral inside the circle is a square.

Ferris Wheel The original Ferris wheel was built in 1893 by Pittsburgh, Pennsylvania, bridge builder George W. Ferris.

The Ferris wheel was originally built for the 1893 World’s Fair in Chicago and was later reconstructed for the 1904 World’s Fair in St. Louis. It had a maximum height of

264 feet and a wheel diameter of 250 feet. Find an equation for the wheel if the center of the wheel is on the y-axis.

Ferris Wheel In 2008, the Singapore Flyer opened as the world’s largest Ferris wheel. It has a maximum height of 165 meters and a diameter of 150 meters, with one full

rotation taking approximately 30 minutes. Find an equation for the wheel if the center of the wheel is on the y-axis.

 Weather Satellites Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation $x^2+y^2+2x+4y-4091=0$
. A weather satellite circles 0.6 unit above Earth with the center of its circular orbit at the center of Earth. Find the equation for the orbit of the satellite on this map.

The tangent line to a circle may be defined as the line that intersects the circle in a single point, called the point of tangency. See the figure. 

If the equation of the circle is $x^2+y^2=r^2$ and the equation of the tangent line is y = mx + b, show that:

(a) $r^2(1+m^2)=b^2$ 

[Hint: The quadratic equation $x^2+{(mx+b)}^2=r^2$ has exactly one solution.]

(b) The point of tangency is $(\frac{-r^{2}m}{b},\frac{r^2}{b})$.

(c) The tangent line is perpendicular to the line containing the center of the circle and the point of tangency.

The Greek Method The Greek method for finding the equation of the tangent line to a circle uses the fact that at any point on a circle the lines containing the center and the tangent line are perpendicular (see Problem 52). Use this method to find an equation of the tangent line to the circle

x² + y² = 9 at the point $(1 ,2\sqrt{2).}$

Use the Greek method described in Problem 53 to find an equation of the tangent line to the circle

x² + y² - 4x + 6y + 4 = 0 at the point $\left(3,2\sqrt{2}-3\right)$

Refer to Problem 52. The line x - 2y + 4 = 0 is tangent to a circle at (0, 2). The line 

 y=2x- 7 is tangent to the same

circle at (3, -1). Find the center of the circle.

Find an equation of the line containing the centers of the two circles $$\begin{gathered} x^2+y^2-4x+6y+4 =0 and \\ {x}^2+y^2+6x+4y+9=0 \end{gathered}$$

If a circle of radius 2 is made to roll along the x-axis, what is an equation for the path of the center of the circle?

If the circumference of a circle is $6\mathrm{\pi }$, what is its radius?

Which of the following equations might have the graph shown? (More than one answer is possible.) 

$$\begin{gathered} \left(a\right) (x-2{)}^2+(y+3{)}^2=13 \\ \left(b\right) (x-2{)}^2+(y-2{)}^2=8 \\ \left(c\right) (x-2{)}^2+(y-3{)}^2=13 \\ \left(d\right) (x+2{)}^2+(y-2{)}^2=8 \\ \left(e\right) x^2+y^2-4x-9y=0 \\ \left(f\right) x^2+y^2+4x-2y=0 \\ \left(g\right) x^2+y^2-9x-4y=0 \\ \left(h\right) x^2+y^2-4x-4y=4 \end{gathered}$$

Which of the following equations might have the graph shown? (More than one answer is possible.) 

$$\begin{gathered} \left(a\right) (x-2{)}^2+y^2=3 \\ (b) (x+2{)}^2+y^2=3 \\ \left(c\right) x^2+(y-2{)}^2=3 \\ (d) (x+2{)}^2+y^2=4 \\ \left(e\right) x^2+y^2+10x+16=0 \\ \left(f\right) x^2+y^2+10x-2y=1 \\ \left(g\right) x^2+y^2+9x+10=0 \\ \left(h\right) x^2+y^2-9x-10=0 \end{gathered}$$

Explain how the center and radius of a circle can be used to graph the circle.

What Went Wrong? A student stated that the center and radius of the graph whose equation is (x + 3)²+ (y - 2)²= 16 are

(3, -2) and 4, respectively. Why is this incorrect?

Open the “Circle: the role of the center” applet. Place the cursor on the center of the circle and hold the mouse button. Drag the center around the Cartesian plane and note how the equation of the circle changes.

(a) What is the radius of the circle?

(b) Draw a circle whose center is at (1, 3). What is the equation of the circle?

(c) Draw a circle whose center is at ( -1, 3). What is the equation of the circle?

(d) Draw a circle whose center is at (-1, -3). What is the equation of the circle?

(e) Draw a circle whose center is at (1, -3). What is the equation of the circle?

(f) Write a few sentences explaining the role the center of the circle plays in the equation of the circle.

Radius of a Circle Open the “Circle: the role of the radius” applet. Place the cursor on point B, press and hold the mouse button. Drag B around the Cartesian plane. 

(a) What is the center of the circle? 

(b) Move B to a point in the Cartesian plane directly above the center such that the radius of the circle is 5. 

(c) Move B to a point in the Cartesian plane such that the radius of the circle is 4. 

(d) Move B to a point in the Cartesian plane such that the radius of the circle is 3. 

(e) Find the coordinates of two points with integer coordinates in the fourth quadrant on the circle that result in a circle of radius 5 with center equal to that found in part (a). 

(f) Use the concept of symmetry about the center, vertical line through the center of the circle, and horizontal line through the center of the circle to find three other points with integer coordinates in the other three quadrants that lie on the circle of radius 5 with center equal to that found in part (a).

Find the following for each pair of points:

(a) The distance between the points.

(b) The midpoint of the line segment connecting the points.

(c) The slope of the line containing the points.

(d) Then interpret the slope found in part (c). 

$\left(0,0\right);\left(4,2\right)$.

Find the following for each pair of points:

(a) The distance between the points.

(b) The midpoint of the line segment connecting the points.

(c) The slope of the line containing the points.

(d) Then interpret the slope found in part (c). 

$\left(1,-1\right);\left(-2,3\right)$.

Find the following for each pair of points:

(a) The distance between the points.

(b) The midpoint of the line segment connecting the points.

(c) The slope of the line containing the points.

(d) Then interpret the slope found in part (c). 

$\left(4,-4\right);\left(4,8\right)$.

Find the following for each pair of points:

(a) The distance between the points.

(b) The midpoint of the line segment connecting the points.

(c) The slope of the line containing the points.

(d) Then interpret the slope found in part (c). 

$\left(-2,-1\right);\left(3,-1\right)$.

List the intercepts of the following graph. 

Graph $y=-x^2+15$ using a graphing utility. Create a table of values to determine a good initial viewing window. Use a graphing utility to approximate the intercepts. 

Determine the intercepts and graph each equation by hand by plotting points. Verify your results using a graphing utility. Label the intercepts on the graph. 

$2x-3y=6$.

Determine the intercepts and graph each equation by hand by plotting points. Verify your results using a graphing utility. Label the intercepts on the graph.  
$y=x^2-9$.