91Ó°ÊÓ

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=-8 y^{2}$$

A polar equation of a conic is given. (a) Show that the conic is a parabola and sketch its graph. (b) Find the vertex and directrix and indicate them on the graph. $$r=\frac{4}{1-\sin \theta}$$

A polar equation of a conic is given. (a) Show that the conic is a parabola and sketch its graph. (b) Find the vertex and directrix and indicate them on the graph. $$r=\frac{3}{2+2 \sin \theta}$$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(xy\)-term. (c) Sketch the graph. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}=16$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$x^{2}-2 y^{2}=3$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$9 x^{2}+4 y^{2}=1$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=\frac{1}{2} y^{2}$$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(xy\)-term. (c) Sketch the graph. $$11 x^{2}-24 x y+4 y^{2}+20=0$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x^{2}+6 y=0$$

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