91Ó°ÊÓ

Use a graphing device to graph the conic. $$9 x^{2}+36=y^{2}+36 x+6 y$$

Use a graphing device to graph the conic. $$x^{2}-4 y^{2}+4 x+8 y=0$$

Find an equation for the hyperbola that satisfies the given conditions. Vertices: \((0, \pm 6),\) hyperbola passes through \((-5,9)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix has \(y\) -intercept 6

Find an equation for the ellipse that satisfies the given conditions. Foci: \((\pm 5,0),\) length of major axis: 12

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Opens upward with focus 5 units from the vertex

Determine what the value of \(F\) must be if the graph of the equation $$4 x^{2}+y^{2}+4(x-2 y)+F=0$$ is (a) an ellipse, (b) a single point, or (c) the empty set.

Find an equation for the ellipse that satisfies the given conditions. Endpoints of major axis: \((\pm 10,0),\) distance between foci: 6

Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance \(d(P, Q)\) and \(d\left(P^{\prime}, Q^{\prime}\right)\) where \(P^{\prime}\) and \(Q^{\prime}\) are the images of \(P\) and \(Q\) under a rotation of axes.

Find an equation for the hyperbola that satisfies the given conditions. Asymptotes: \(y=\pm x,\) hyperbola passes through \((5,3)\)