Chapter 10

Fill in the blanks. \(\left\\{\begin{array}{l}4 x^{2}+6 y^{2}=24 \\ 9 x^{2}-y^{2}=9\end{array}\right.\) is a _____ of two nonlinear equations.

Fill in the blanks. The curves formed by the intersection of a plane with an infinite right- circular cone are called _____ ______.

Fill in the blanks. The graph of \(2 x+y=10\) is a _____ and the graph of \(x^{2}+y^{2}=25\) is a _____.

A ______ is the set of all points in a plane for which the difference of the distances from two fixed points is a constant.

Fill in the blanks. An ________ is the set of all points in a plane for which the sum of the distances from two fixed points is a constant.

Fill in the blanks. When solving a system by graphing, it is often difficult to determine the coordinates of the points of _____ of the graphs.

Fill in the blanks. \(A\)____ is the set of all points in a plane that are a fixed distance from a fixed point called its center. The fixed distance is called the _____ .

Fill in the blanks. Two algebraic methods for solving systems of nonlinear equations are the _____ method and the _____ method.

Fill in the blanks. A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed _____.

Fill in the blanks. A _____ is a line that intersects a circle at two points.

Fill in the blanks. The line segment joining the vertices of an ellipse is called the ________ axis of the ellipse.

A. Write the standard form of the equation of a circle. B. Write the standard form of the equation of a circle with the center at the origin.

Fill in the blanks. A _____ is a line that intersects a circle at one point.

Fill in the blanks. To write \(9 x^{2}-4 y^{2}=36\) in ______ form, we divide both sides by 36

Fill in the blanks. The midpoint of the major axis of an ellipse is the ________ of the ellipse.

a. A line can intersect an ellipse in at most _____ points. b. An ellipse can intersect a parabola in at most _____ points. c. An ellipse can intersect a circle in at most _____ points. d. A hyperbola can intersect a circle in at most _____ points.

Write the standard form of the equation of a hyperbola centered at the origin that opens left and right.

Write the standard form of the equation of an ellipse centered at the origin and symmetric to both axes.

Determine whether \((1,-1)\) is a solution of the system: $$\left\\{\begin{array}{l}2 x+y-1=0 \\ x^{2}-y^{2}=3\end{array}\right.$$

Write the standard form of the equation of a hyperbola centered at \((h, k)\) that opens up and down.

Write the standard form of the equation of a horizontal or vertical ellipse centered at \((h, k)\)

Find the solutions of the system \(\left\\{\begin{array}{l}x^{2}+4 y^{2}=25 \\\ x^{2}-2 y^{2}=1\end{array}\right.\) on the right.

Find the \(x\) -and the \(y\) -intercepts of the graph of \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

A. What is the standard form of the equation of a parabola opening upward or downward? B. What is the standard form of the equation of a parabola opening to the right or left?

Find a substitution equation that can be used to solve the system: $$\left\\{\begin{array}{l}x^{2}+y^{2}=9 \\ 2 x-y=3\end{array}\right.$$

Consider the system: \(\left\\{\begin{array}{l}6 x^{2}+y^{2}=9 \\ 3 x^{2}+4 y^{2}=36\end{array}\right.\) a. If the \(y^{2}\) -terms are to be eliminated, by what should the first equation be multiplied? b. If the \(x^{2}\) -terms are to be eliminated, by what should the second equation be multiplied?

Determine whether the graph of each equation is a circle or a parabola. A. \(x^{2}+y^{2}-6 x+8 y-10=0\) B. \(y^{2}-2 x+3 y-9=0\) C. \(x^{2}+5 x-y=0\) D. \(x^{2}+12 x+y^{2}=0\)

Suppose you begin to solve the system \(\left\\{\begin{array}{l}x^{2}+y^{2}=10 \\\ 4 x^{2}+y^{2}=13\end{array}\right.\) and find that \(x\) is \(\pm 1 .\) Use the first equation to find the corresponding \(y\) -values for \(x=1\) and \(x=-1 .\) State the solutions as ordered pairs.

$$\begin{array}{|r|r|}\hline x & y \\\\\hline-2 & \\\\\hline 5 & \\\\\hline\end{array}$$ a. Fill in the blank: An equation of the form \(x y=k,\) where \(k \neq 0,\) has a graph that is a that does not intersect either the \(x\) -axis or the \(y\) -axis.

Draw a parabola using the given facts. Opens right Passes through \((-2,1)\) Vertex \((-3,2)\) \(x\) -intercept \((1,0)\)

Find two points on the graph of \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\) by letting \(x=2\) and finding the corresponding values of \(y .\)

Complete each solution to solve the system. Solve: \(\left\\{\begin{array}{l}x^{2}+y^{2}=5 \\ y=2 x\end{array}\right.\) $$ \begin{aligned} x^{2}+y^{2} &=5 \\ x^{2}+(\square)^{2} &=5 \\ x^{2}+4 x^{2} &=\square\\\ \square x^{2} &=5 \\ x^{2} &=\square \end{aligned} $$ $$ x=1 \quad \text { or } \quad x=-1 $$ If \(x=1,\) then \(y=2(\square)=2\) If \(x=-1,\) then \(y=2(\square)=-2\) The solutions are \((1,2)\) and \((-1, \square)\)

Divide both sides of the equation by 100 and write the equation in standard form: $$100(x+1)^{2}-25(y-5)^{2}=100$$

Find \(h, k,\) and \(r:(x-6)^{2}+(y+2)^{2}=9\)

Divide both sides of the equation by 64 and write the equation in standard form: $$ 4(x-1)^{2}+64(y+5)^{2}=64 $$

Determine whether the graph of the equation will be a circle, a parabola, an ellipse, or a hyperbola. a. \(x^{2}+y^{2}=10\) b. \(9 y^{2}-16 x^{2}=144\) c. \(x=y^{2}-3 y+6\) d. \(4 x^{2}+25 y^{2}=100\)

A. Find \(a, h,\) and \(k: y=6(x-5)^{2}-9\) B. Find \(a, h,\) and \(k: x=-3(y+2)^{2}+1\)

Determine whether the graph of each equation is a circle, a parabola, or an ellipse. a. \(x=y^{2}-2 y+10\) b. \(\frac{x^{2}}{49}+\frac{y^{2}}{64}=1\) c. \((x-3)^{2}+(y+4)^{2}=25\) d. \(2(x-1)^{2}+8(y+5)^{2}=32\)

Find \(h, k, a,\) and \(b: \frac{(x-5)^{2}}{25}-\frac{(y+11)^{2}}{36}=1\)

Find the center and radius of each circle and graph it. $$ x^{2}+y^{2}=9 $$

Find \(h, k, a,\) and \(b: \frac{(x+8)^{2}}{100}+\frac{(y-6)^{2}}{144}=1\)

Solve each system of equations by graphing. See Example 1. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=16 \\ y-x=-4 \end{array}\right. $$

Write each denominator in the equation \(\frac{x^{2}}{36}-\frac{y^{2}}{81}=1\) as the square of a number.

Find the center and radius of each circle and graph it. $$ x^{2}+y^{2}=16 $$

Write each denominator in the equation \(\frac{x^{2}}{81}+\frac{y^{2}}{49}=1\) as the square of a number.

Find the center and radius of each circle and graph it. $$ x^{2}+(y+3)^{2}=1 $$

Graph each equation. \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\)

Find the center and radius of each circle and graph it. $$ (x+4)^{2}+y^{2}=1 $$

Graph each equation. \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\)

Find the center and radius of each circle and graph it. $$ (x+3)^{2}+(y-1)^{2}=16 $$