Chapter 11

Suppose we want to graph an equation in \(x\) and \(y .\) (a) If we replace \(x\) by \(x-3,\) the graph of the equation is shifted to the ___________ by 3 units, If we replace \(x\) by \(x+3,\) the graph of the equation is shifted to the __________ by 3 units. (b) If we replace \(y\) by \(y-1,\) the graph of the equation is shifted ____________ by 1 unit. If we replace \(y\) by \(y+1\) the graph of the equation is shifted _________ by 1 unit.

A hyperbola is the set of all points in the plane for which the _____________ of the distances from two fixed points \(F_{1}\) and \(F_{2}\) is constant. The points \(F_{1}\) and \(F_{2}\) are called the ______________ of the hyperbola.

An ellipse is the set of all points in the plane for which the _________ of the distances from two fixed points \(F_{1}\) and \(F_{2}\) is constant. The points \(F_{1}\) and \(F_{2}\) are called the ________ of ellipse.

A parabola is the set of all points in the plane that are equidistant from a fixed point called the ____ and a fixed line called the ____ of the parabola.

The graph of the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a>b>0\) is an ellipse with vertices (________ , ________ )and (_______ ,________ ) and foci \((\pm c, 0),\) where \(c=\) ________ . So the graph of \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\) is an ellipse with vertices (_______ , _______ ) and (_______,_______) and foci (_______ , _______ ) and (_______,_______).

The graph of the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with \(a>0, b>0\) is a hyperbola with _______________ (horizontal/vertical) transverse axis, vertices (___, ___) and (___, ___) and foci \((\pm c, 0),\) where \(c=\) _______________ . So the graph of \(\frac{x^{2}}{4^{2}}-\frac{y^{2}}{3^{2}}=1\) is a hyperbola with vertices (___, ___) and (___, ___) and foci (___, ___) and (___, ___).

The graphs of \(x^{2}=12 y\) and \((x-3)^{2}=12(y-1)\) are given. Label the focus, directrix, and vertex on each parabola. (Graph cannot copy)

The graph of the equation \(x^{2}=4 p y\) is a parabola with focus \(F\) (____, ____) and directrix \(y=\) _____. So the graph of \(x^{2}=12 y\) is a parabola with focus \(F\) (_____, ____) and directrix y= ____.

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(1,1), \quad \phi=45^{\circ}$$

The graph of the equation \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) with \(a>b>0\) is an ellipse with vertices (________ , _______) and (_______ , _______) and foci \((0, \pm c),\) where \(c=\) ________. \text { So the graph of } \frac{x^{2}}{4^{2}}+\frac{y^{2}}{5^{2}}=1 is an ellipse with vertices (_______ , _______) and (_______ , _______)and foci (_______ , _______) and (_______ , _______).

The graphs of \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\) and \(\frac{(x-3)^{2}}{5^{2}}+\frac{(y-1)^{2}}{4^{2}}=1\) are given. Label the vertices and foci on each ellipse. (Graph cannot copy)

The graph of the equation \(y^{2}=4 p x\) is a parabola with focus \(F\) (____, ____) and directrix \(x=\) ____. So the graph of \(y^{2}=12 x\) is a parabola with focus \(F\) (____, ____) and directrix x= ____.

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity $$\$fTrueac{2}{3},$$ directrix $x=3$$

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(-2,1), \quad \phi=30^{\circ}$$

Label the vertices and foci on the graphs given for the ellipses in Exercises 2 and 3 . (a) \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\) (b) \(\frac{x^{2}}{4^{2}}+\frac{y^{2}}{5^{2}}=1\) (GRAPH CAN'T COPY)

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(\frac{4}{3},\) directrix \(x=-3\)

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(3,-\sqrt{3}), \quad \phi=60^{\circ}$$

An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Parabola, directrix \(y=2\)

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(2,0), \quad \phi=15^{\circ}$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{1}{2},\) directrix \(y=-4\)

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(0,2), \quad \phi=55^{\circ}$$

An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{x^{2}}{9}+\frac{(y+5)^{2}}{25}=1$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(4,\) directrix \(r=5 \sec \theta\)

Determine the XY-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(\sqrt{2}, 4 \sqrt{2}), \quad \phi=45^{\circ}$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.6,\) directrix \(r=2 \csc \theta\)

An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. $$\frac{x^{2}}{4}-\frac{y^{2}}{16}=1$$

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}-3 y^{2}=4, \quad \phi=60^{\circ}$$

An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{(x+5)^{2}}{16}+\frac{(y-1)^{2}}{4}=1$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Parabola, vertex at \((5, \pi / 2)\)

An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$

An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. $$\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$$

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$y=(x-1)^{2}, \quad \phi=45^{\circ}$$

An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{(x+1)^{2}}{36}+\frac{(y+1)^{2}}{64}=1$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.4,\) vertex at \((2,0)\)

An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$$

An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. $$\frac{y^{2}}{36}-\frac{x^{2}}{4}=1$$

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}-y^{2}=2 y, \quad \phi=\cos ^{-1} \frac{3}{5}$$

An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$4 x^{2}+25 y^{2}-50 y=75$$

An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. $$x^{2}=8 y$$

An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{x^{2}}{36}+\frac{y^{2}}{81}=1$$

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}+2 y^{2}=16, \quad \phi=\sin ^{-1} \frac{3}{5}$$

An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$9 x^{2}-54 x+y^{2}+2 y+46=0$$

An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. $$x^{2}=-4 y$$

An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{x^{2}}{4}+y^{2}=1$$

An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. $$\frac{x^{2}}{9}-\frac{y^{2}}{64}=1$$

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}+2 \sqrt{3} x y-y^{2}=4, \quad \phi=30^{\circ}$$

An equation of a parabola is given. (a) Find the vertex, focus, and directrix of the parabola. (b) Sketch a graph showing the parabola and its directrix. $$(x-3)^{2}=8(y+1)$$

An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. $$y^{2}=-24 x$$

An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. $$\frac{x^{2}}{49}+\frac{y^{2}}{25}=1$$