Use your answers from the given equations a. A circle: $x=h+r \cos t, y=k+r \sin t$ b. An ellipse: $x=h+a \cos t, y=k+b \sin t$ c. A hyperbola: $x=h+a \sec t, y=k+b \tan t$ to find a set of parametric equations for the conic section or the line. Ellipse: Center: $(4,-1)$; Vertices: 5 units above and below the center; Endpoints of Minor Axis: 3 units to the left and right of the center

a. Rewrite the equation in a rotated x'y' system without an x'y'. b. Express the equation involving x' and y' in the standard form of a conic section. c. Use the rotated system to graph the equation.

$$x^2+x y+y^2-1=0$$

Use a graphing utility to graph the equation.

$$3 x^2-6 x y+3 y^2+10 x-8 y-2=0$$

Graph the ellipse and give the location of its foci.

$$\dfrac{(x-1)^2}{2}+\dfrac{(y+3)^2}{5}=1$$

Use a graphing utility to graph the equation.

$$3 x^2+4 x y+6 y^2-7=0$$

Determine whether the statement makes sense or does not make sense, and explain your reasoning. I graphed a conic in the form $r=\dfrac{e p}{1-e \cos \theta}$ that was symmetric with respect to the $y$-axis.

Use your answers from the given equations a. A circle: $x=h+r \cos t, y=k+r \sin t$ b. An ellipse: $x=h+a \cos t, y=k+b \sin t$ c. A hyperbola: $x=h+a \sec t, y=k+b \tan t$ to find a set of parametric equations for the conic section or the line. Ellipse: Center: $(-2,3)$; Vertices: 5 units to the left and right of the center; Endpoints of Minor Axis: 2 units above and below the center

Convert the equation to standard form by completing the square on $x$ or $y$. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.

$$x^2+6 x-4 y+1=0$$

Convert the equation to standard form by completing the square on $x$ and $y$. Then graph the hyperbola. Find the foci and find the equations of the asymptotes.

$$4 x^2-9 y^2-16 x+54 y-101=0$$

a. Rewrite the equation in a rotated x'y' system without an x'y'. b. Express the equation involving x' and y' in the standard form of a conic section. c. Use the rotated system to graph the equation.

$$x y-4=0$$

Convert the equation to standard form by completing the square on $x$ or $y$. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.

$$y^2-2 y-8 x+1=0$$

Identify the conic defined by the equation without using a rotation of axes.

$$x^2-2 x y+3 y^2+2 x+4 y-1=0$$

Convert the equation to standard form by completing the square on $x$ and $y$. Then graph the hyperbola. Find the foci and find the equations of the asymptotes.

$$9 y^2-4 x^2-18 y+24 x-63=0$$

Determine whether the statement makes sense or does not make sense, and explain your reasoning. Eccentricity and polar coordinates enable me to see that ellipses, hyperbolas, and parabolas are a unified group of interrelated curves.