Chapter 2

\(83-86\) Complete the graph using the given symmetry property. Symmetric with respect to the \(y\) -axis

\(87-90\) . Sketch the region given by the set. $$ \left\\{(x, y) | x^{2}+y^{2} \leq 1\right\\} $$

\(87-90\) . Sketch the region given by the set. $$ \left\\{(x, y) | x^{2}+y^{2}>4\right\\} $$

\(87-90\) . Sketch the region given by the set. $$ \left\\{(x, y) | 1 \leq x^{2}+y^{2}<9\right\\} $$

\(87-90\) . Sketch the region given by the set. $$ \left\\{(x, y) | 2 x

Find the area of the region that lies outside the circle \(x^{2}+y^{2}=4\) but inside the circle $$ x^{2}+y^{2}-4 y-12=0 $$

Sketch the region in the coordinate plane that satisfies both the inequalities \(x^{2}+y^{2} \leq 9\) and \(y \geq|x| .\) What is the area of this region?

Orbit of a Satellite A satellite is in orbit around the moon. A coordinate plane containing the orbit is set up with the center of the moon at the origin, as shown in the graph below, with distances measured in megameters (Mm). The equation of the satellite's orbit is $$\frac{(x-3)^{2}}{25}+\frac{y^{2}}{16}=1$$ (a) From the graph, determine the closest and the farthest that the satellite gets to the center of the moon. (b) There are two points in the orbit with \(y\) -coordinates 2 Find the \(x\) -coordinates of these points, and determine their distances to the center of the moon.

Circle, Point, or Empty Set? 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 in which the equation does represent a circle, find its center and radius.

(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. $$\begin{aligned} \text { (i) }(x-2)^{2}+(y-1)^{2} &=9 \\\\(x-6)^{2}+(y-4)^{2} &=16 \\ \text { (ii) } x^{2}+(y-2)^{2}=4 & \\\\(x-5)^{2}+(y-14)^{2} &=9 \\\ \text { (iii) }(x-3)^{2}+(y+1)^{2} &=1 \\\\(x-2)^{2}+(y-2)^{2} &=25 \end{aligned}$$ (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.