91Ó°ÊÓ

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 10),\) vertices: \((0, \pm 8)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(y=\frac{1}{10}\)

Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Foci: \((\pm 4,0),\) vertices: \((\pm 5,0)\)

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 2),\) vertices: \((0, \pm 1)\)

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 conic section with the given properties. The parabola with vertex \(V(-3,5)\) and directrix \(y=2\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(x=-\frac{1}{8}\)

Find an equation for the conic section with the given properties. The parabola with focus \(F(1,3)\) and directrix \(x=3\)

Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Foci: \((0, \pm 3),\) vertices: \((0, \pm 5)\)

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((\pm 6,0),\) vertices: \((\pm 2,0)\)