Stewart Precalc 6e Section 1.8: Coordinate Geometry

1 Application - Midpoint Endpoint · Level 2

Plot the points \(M(6, 8)\) and \(A(2, 3)\) on a coordinate plane. If \(M\) is the midpoint of segment \(A B\), find the coordinates of \(B\). Write a brief description of the steps you took to find \(B\), and your reasons for taking them.

2 Application - Parallelogram Construction · Level 3

Plot the points \(P(0, 3)\), \(Q(2, 2)\), and \(R(5, 3)\) on a coordinate plane. Where should the point \(S\) be located so that \(P Q R S\) is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

3 Application - General Circle Equation · Level 4

Complete the squares in the general equation \(x^2 + a x + y^2 + b y + c = 0\) and simplify the result as much as possible. Under what conditions on the coefficients \(a\), \(b\), and \(c\) does this equation represent a circle? A single point? The empty set? In the case that the equation does represent a circle, find its center and radius.

4 Exercise - Circle Intersection · Level 3

Do the Circles Intersect?

(a) Find the radius of each circle in the pair and the distance between their centers; then use this information to determine whether the circles intersect. (i) \((x - 2)^2 + (y - 1)^2 = 9\); \((x - 6)^2 + (y - 4)^2 = 16\) (ii) \(x^2 + (y - 2)^2 = 4\); \((x - 5)^2 + (y - 14)^2 = 9\) (iii) \((x - 3)^2 + (y + 1)^2 = 1\); \((x - 2)^2 + (y - 2)^2 = 25\)

Answer for 4(a)

(b) How can you tell, just by knowing the radii of two circles and the distance between their centers, whether the circles intersect? Write a short paragraph describing how you would decide this and draw graphs to illustrate your answer.

Answer for 4(b)

Enter your answer directly below each part above.

5 Exercise - Symmetry · Level 2

Making a Graph Symmetric. The graph shown in the figure is not symmetric about the \(x\)-axis, the \(y\)-axis, or the origin. Add more line segments to the graph so that it exhibits the indicated symmetry. In each case, add as little as possible.

(a) Symmetry about the \(x\)-axis

Answer for 5(a)

(b) Symmetry about the \(y\)-axis

Answer for 5(b)

(c) Symmetry about the origin

Answer for 5(c)

Enter your answer directly below each part above.

6 Example - Graphing Regions in the Coordinate Plane · Level 2

Describe and sketch the regions given by each set. (a) All points \((x, y)\) with \(x \geq 0\). (b) All points \((x, y)\) with \(y = 1\). (c) All points \((x, y)\) with \(|y| < 1\).

7 Example - Distance Formula · Level 2

Which of the points \(P(1, -2)\) or \(Q(8, 9)\) is closer to the point \(A(5, 3)\)?

8 Example - Midpoint Formula · Level 3

Show that the quadrilateral with vertices \(P(1, 2)\), \(Q(4, 4)\), \(R(5, 9)\), and \(S(2, 7)\) is a parallelogram by proving that its two diagonals bisect each other.