91Ó°ÊÓ

Let \(P_{3}\) be the midpoint of the line segment between \(P_{1}(-3,4,1)\) and \(P_{2}(-5,8,3) .\) Find the coordinates of the midpoint of the line segment: (a) between \(P_{1}\) and \(P_{3}\) (b) between \(P_{3}\) and \(P_{2}\).

Find an equation of parabola that satisfies the given conditions. Vertex \((0,0),\) directrix \(y=-\frac{7}{4}\)

Find an equation of the ellipse that satisfies the given conditions. Endpoints of minor axis \((0,5),(0,-1),\) one focus (6,2)

In Problems \(21-44,\) find an equation of the hyperbola that satisfies the given conditions. Vertices (±2,0) , passing through \((2 \sqrt{3}, 4)\)

Sketch the graph of the given equation. $$ x^{2}+y^{2}+z^{2}=9 $$

Sketch the graph of the given equation. $$ x^{2}+y^{2}+(z-3)^{2}=16 $$

Find an equation of the ellipse that satisfies the given conditions. Endpoints of major axis \((2,4),(13,4),\) one focus (4,4)

Find an equation of parabola that satisfies the given conditions. Vertex \((0,0),\) directrix \(x=6\)

In Problems \(21-44,\) find an equation of the hyperbola that satisfies the given conditions. Vertices \((0,\pm 3),\) passing through \(\left(\frac{16}{5}, 5\right)\)

In Problems 41-44, find a function \(f\) that defines the indicated half- ellipse. Give the domain of each function. The equations are from Problems 1,3,9, and 12 $$ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \text { ; lower half-ellipse } $$