Chapter 13

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ 5(x-3)^{2}+5(y-3)^{2}=30 $$ \((3,3) ; r=\sqrt{6}\)

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ x^{2}-6 x+4 y^{2}+5=0 $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ (y+4)^{2}=-8(x+2) $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. x^{2}+y^{2}-6 x-10 y+30=0

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 16 x^{2}+9 y^{2}+36 y-108=0 $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ (y-3)^{2}=8(x-1) $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ x^{2}+y^{2}-6 x-10 y+30=0 \quad(3,5), r=2 $$

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 9 x^{2}-72 x+2 y^{2}+4 y+128=0 $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ x^{2}-2 x-4 y+9=0 $$

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 5 x^{2}+10 x+16 y^{2}+160 y+325=0 $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ x^{2}+4 x-8 y-4=0 $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ x^{2}+y^{2}+10 x+14 y+73=0 \quad(-5,-7), r=1 $$

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 2 x^{2}+12 x+11 y^{2}-88 y+172=0 $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ x^{2}+6 x+8 y+1=0 $$

Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse, and sketch its graph. See answer section. $$ 9 x^{2}+72 x+y^{2}+6 y+135=0 $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ x^{2}-4 x+4 y-4=0 $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ x^{2}+y^{2}+7 x-2=0 \quad\left(-\frac{7}{2}, 0\right) ; r=\frac{\sqrt{57}}{2} $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}-2 y+12 x-35=0 $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ x^{2}+y^{2}-5 y-1=0 \quad\left(0, \frac{5}{2}\right): r=\frac{\sqrt{29}}{2} $$

Find an equation of the ellipse that satisfies the oiven conditions. Vertices \((\pm 4,0)\), foci \((\pm 2,0) \quad 3 x^{2}+4 y^{2}=48\)

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}+4 y+8 x-4=0 $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ x^{2}+y^{2}-4 x+2 y=0 \quad(2,-1), r=\sqrt{5} $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ x^{2}+y^{2}=8 \quad(0,0), r=2 \sqrt{2} $$

For Problems \(1-30\), find the vertex, focus, and directrix of the given parabola and sketch its graph. $$ y^{2}-6 y-12 x+21=0 $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ 4 x^{2}+4 y^{2}=1 \quad(0,0), r=\frac{1}{2} $$

Find an equation of the ellipse that satisfies the oiven conditions. Vertices \((0, \pm 5)\), length of minor axis is 4 $$ 25 x^{2}+4 y^{2}=100 $$

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. $$ \text { Focus }\left(0,-\frac{1}{2}\right) \text {, directrix } y=\frac{1}{2} \quad x^{2}=-2 y $$

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. $$ \text { Focus }(-1,0) \text {, directrix } x=1 \quad y^{2}=-4 x $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. Find the equation of the line that is tangent to the circle \(x^{2}+y^{2}-2 x+3 y-12=0\) at the point \((4,1)\) \(6 x+5 y=29\)

Find an equation of the ellipse that satisfies the oiven conditions. $$ \text { Foci }(\pm 1,0) \text {, length of minor axis is } 2 \quad x^{2}+2 y^{2}=2 $$

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. $$ \text { Focus }(5,0) \text {, directrix } x=1 \quad y^{2}-8 x+24=0 $$

For Problems \(15-32\), find the center and the length of a radius of each of the circles. Find the equation of the line that is tangent to the circle \(x^{2}+y^{2}+4 x-6 y-4=0\) \(x-4 y=3\) at the point \((-1,-1)\)

Find the equation of the circle that passes through the origin and has its center at \((-3,-4)\). \(x^{2}+y^{2}+6 x+8 y=0\)

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. $$ \text { Focus }(-3,-1) \text {, directrix } y=7 \quad x^{2}+6 x+16 y-39=0 $$

Find the equation of the circle that has its center at \((-2,-3)\) and is tangent to the line \(x+y=-3\).

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. $$ \text { Focus }(-4,5) \text {, directrix } x=0 \quad y^{2}-10 y+8 x+41=0 $$

The point \((-1,4)\) is the midpoint of a chord of a circle whose equation is \(x^{2}+y^{2}+8 x+4 y-30=0\). Find the equation of the chord.

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. $$ \text { Focus }(5,-2) \text {, directrix } x=-1 \quad y^{2}+4 y-12 x+28=0 $$

Find the equation of the circle that is tangent to the line \(3 x-4 y=-26\) at the point \((-2,5)\) and passes through the point \((5,-2) . \quad x^{2}+y^{2}-2 x-2 y-23=0\)

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. Vertex \((0,0)\), symmetric with respect to the \(x\) axis, and contains the point \((-3,5) \quad 3 y^{2}=-25 x, y^{2}=-\frac{25}{3} x\)

Find the equation of the circle that is tangent to the line \(3 x-4 y=-26\) at the point \((-2,5)\) and passes through the point \((5,-2) . \quad x^{2}+y^{2}-2 x-2 y-23=0\)

Find the equation of the circle that passes through the three points \((3,0),(6,-9)\) and \((10,-1)\).

What is the graph of the equation \(x^{2}+y^{2}=0\) ? Explain your answer.

Vertex \((0,0)\), focus \(\left(0,-\frac{7}{2}\right) \quad x^{2}=-14 y\)

What is the graph of the equation \(x^{2}+y^{2}=-4\) ? \(\mathrm{Ex}-\) plain your answer.

What type of figure is the graph of the equation \(x^{2}+\) \(6 x+2 y^{2}-20 y+59=0 ?\) Explain your answer.

Your friend claims that the graph of an equation of the form \(x^{2}+y^{2}+D x+E y+F=0\), where \(F=0\), is a circle that passes through the origin. Is she correct? Explain why or why not.

Suppose that someone graphed the equation \(4 x^{2}-16 x\) \(+9 y^{2}+18 y-11=0\) and obtained the graph shown in Figure 13.31. How do you know by looking at the equation that this is an incorrect graph?

By expanding \((x-h)^{2}+(y-k)^{2}=r^{2}\), we obtain \(x^{2}-\) \(2 h x+h^{2}+y^{2}-2 k y+k^{2}-r^{2}=0\). When we compare this result to the form \(x^{2}+y^{2}+D x+E y+F=0\), we see that \(D=-2 h, E=-2 k\), and \(F=h^{2}+k^{2}-r^{2}\). Therefore, solving those equations respectively for \(h, k\), and \(r\), we can find the center and the length of a radius of a circle by using \(h=\frac{D}{-2}, k=\frac{E}{-2}\), and \(r=\sqrt{h^{2}+k^{2}-F}\). Use these relationships to find the center and the length of a radius of each of the following circles: (a) \(x^{2}+y^{2}-2 x-8 y+8=0 \quad(1,4), r=3\) (b) \(x^{2}+y^{2}+4 x-14 y+49=0 \quad(-2,7), r=2\) (c) \(x^{2}+y^{2}+12 x+8 y-12=0 \quad(-6,-4), r=8\) (d) \(x^{2}+y^{2}-16 x+20 y+115=0 \quad(8,-10), r=7\) (e) \(x^{2}+y^{2}-12 x-45=0 \quad(6,0), r=9\) (f) \(x^{2}+y^{2}+14 x=0 \quad(-7,0), r=7\)

Vertex \((-9,1)\), symmetric with respect to the line \(x=-9\), and contains the point \((-8,0)\) $$ x^{2}+18 x+y+80=0 $$