Chapter 11

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.

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.

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 the 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 \(a>0, b>0\) is a hyperbola with 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 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 \(x^{2}=4 p y\) 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 \(y=\) __________________ .

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

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

Determine the \(X Y\)-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 \(c=\) ____________. 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. CAN'T COPY THE GRAPH

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. Hyperbola, eccentricity \(\frac{4}{3},\) directrix \(x=-3\)

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

The graphs of \(\frac{x^{2}}{4^{2}}-\frac{y^{2}}{3^{2}}=1\) and \(\frac{(x-3)^{2}}{4^{2}}-\frac{(y-1)^{2}}{3^{2}}=1\) are given. Label the vertices, foci, and asymptotes on each hyperbola. CAN'T COPY THE GRAPH

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

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$

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

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

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{(x-3)^{2}}{16}+(y+3)^{2}=1$$

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

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{x^{2}}{9}+\frac{(y+5)^{2}}{25}=1$$

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

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

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{(x+2)^{2}}{4}+y^{2}=1$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{x^{2}}{4}-\frac{y^{2}}{16}=1$$

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

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

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$

Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$(x-3)^{2}=8(y+1)$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$$

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 \((2,0)\)

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

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$$

Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$(y+5)^{2}=-6 x+12$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$y^{2}-\frac{x^{2}}{25}=1$$

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

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$9 x^{2}+4 y^{2}=36$$

Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$-4\left(x+\frac{1}{2}\right)^{2}=y$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x^{2}=9 y$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{x^{2}}{2}-y^{2}=1$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$4 x^{2}+25 y^{2}=100$$

Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$y^{2}=16 x-8$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x^{2}=y$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$x^{2}-y^{2}=1$$

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

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$x^{2}+4 y^{2}=16$$