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Math Topics

College Algebra

Conic Sections

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1
The Parabola

Consider the parabola with vertex at the origin defined by the equation \(y=\dfrac{1}{6}x^2\)

Find the coordinates of the focus.

Find the equations of the directrix and the axis of symmetry.

Find the value(s) of \(a\) for which the point \((a,4)\) is on the parabola.

Sketch the parabola, and indicate the focus and the directrix.

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.

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

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

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2
The Ellipse

Consider the ellipse that is centered at the origin and defined by the equation \(16x^2+25y^2=400\)

Write the equation of the ellipse in standard form, and determine the orientation of the major axis.

Find the coordinates of the vertices and the foci.

Sketch the ellipse, and indicate the vertices and foci.

The orbit of Halleyâ€™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 Halleyâ€™s comet, using the origin as the center of the ellipse and a segment of the \(x\)-axis as the major axis.

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3
The Hyperbola

Consider the hyperbola that is centered at the origin and defined by the equation \(16x^2-9y^2=144\)

Write the equation of the hyperbola in standard form, and determine the orientation of the transverse axis.

Find the equations of the asymptotes.

Sketch the hyperbola, and indicate the vertices, foci, and asymptotes.

Determine the equation in standard form of the hyperbola with center at \((0,0)\), one focus at \((0,4)\), and one vertex at \((0,-1)\). Find the other focus and the other vertex. Sketch a graph of the hyperbola by finding and plotting some additional points that lie on the hyperbola.

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?