Determine the equation in standard form of the parabola with vertex at the origin and directrix \(x=3\). Sketch the parabola, and indicate the focus and the directrix.

Determine the equation in standard form of the parabola with directrix \(y=7\) and focus at \((-3,3)\). Sketch the parabola.

Find the vertex and focus of the parabola \[-4x+3y^2+12y-8=0\] Determine the equation of the directrix and sketch the parabola.

The cross-section of a headlight reflector is in the shape of a parabola. The reflector is \(6\) inches in diameter and \(5\) inches deep, as illustrated in Figure 15.

1. Find an equation of the parabola, using the position of the vertex of the parabola as the origin of your coordinate system.

2. The bulb for the headlight is positioned at the focus. Find the position of the bulb.

Determine the equation in standard form of the ellipse with foci at \((4,1)\) and \((4, -5)\) and one vertex at \((4, 3)\). Sketch the ellipse.

Write the equation in standard form of the ellipse defined by the equation \[5x^2-30x+25y^2+50y+20=0\] Sketch the ellipse.

The orbit of Halley’s comet is elliptical, with the sun at one of the foci. The length of the major axis of the orbit is approximately \(36\) astronomical units (AU), and the length of the minor axis is approximately \(9\) AU \((1 \text{ AU} \approx 92\text{,}600\text{,}000 \text{ miles})\). Find the equation in standard form of the path of Halley’s comet, using the origin as the center of the ellipse and a segment of the \(x\)-axis as the major axis.

The front face of a wire frame sculpture is in the shape of the branches of a hyperbola that opens to the side. The transverse axis of the hyperbola is \(40\) inches long. If the base of the sculpture is \(60\) inches below the transverse axis and one of the asymptotes has a slope of \(\dfrac{3}{2}\), how wide is the sculpture at the base?