Step-by-step solutions to hundreds of conic sections - PlainMath

Recent questions in Conic sections

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asked 2021-08-14

A conic section is said to be circle if the eccentricity e=1

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asked 2021-08-14

Identify and graph the conic whose equation is in attachment.
\(\displaystyle{5}{x}^{{2}}+{4}{x}{y}+{2}{y}^{{2}}-{\frac{{{28}}}{{\sqrt{{5}}}}}{x}-{\frac{{{4}}}{{\sqrt{{5}}}}}{y}+{4}={0}\)

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asked 2021-08-14

Identify each conic using eccentricity.
(a) \(r=\frac{4}{1+3\sin \theta}\)
(b) \(r=\frac{7}{1-3\cos \theta}\)
(c) \(r=\frac{8}{6+5\cos \theta}\)
(d) \(r=\frac{3}{2-3\sin \theta}\)

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asked 2021-08-13

The front (and back) of a greenhouse have the same shape and dimensions shown below. The greenhouse is 40 feet long and the angle at the top of the roof is \(\displaystyle{90}^{{\circ}}\). Determine the volume of the greenhouse in cubic feet. Explain your solution.

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asked 2021-08-13

Your mission is to track incoming meteors to predict whether or not they will strike Earth. Since Earth has a circular cross section, you decide to set up a coordinate system with its origin at Earth's center. The equation of Earth's surface is \(\displaystyle{x}^{{2}}+{y}^{{2}}={40.68}\), where x and y are distances in thousands of kilometers.

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asked 2021-08-13

Find an equation of the conic described.Graph the equation.
Ellipse; center at (0,0); focus at (0,3); vertex at (0, 5)

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asked 2021-08-12

To determine: Find the lateral area of the conical tent.

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asked 2021-08-12

Find the focus, equation of the directrices, lengths of major axis, minor axis and focal diameters, and draw the conic defined by \(\displaystyle{9}{x}^{{2}}+{16}{y}^{{2}}={1}\)

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asked 2021-08-12

Identify what type of conic section is given by the equation below and then find the center, foci, and vertices. If it is a hyperbola, you should also find the asymptotes.
\(\displaystyle{4}{x}^{{2}}-{24}{x}-{4}{y}+{28}={y}^{{2}}\)

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asked 2021-08-12

Find the focus, equation of the directrices, lengths of major axis, minor axis and focal diameters, and draw the conic defined by
\(\displaystyle{4}{x}^{{2}}+{7}{y}^{{2}}={1}\)

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asked 2021-08-12

Solve for the equation in standard form of the following conic sections and graph the curve on a Cartesian plane indicating important points (i.e. vertices and intercepts).The hyperbola given by \(\displaystyle{5}{x}^{{2}}–{4}{y}^{{2}}={20}{x}+{24}{y}+{36}\)

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asked 2021-08-12

Show that for eccentricity equal to one in Equation 13.10 for conic sections, the path is a parabola. Do this by substituting Cartesian coordinates, x and y, for the polar coordinates, r and θ , and showing that it has the general form for a parabola, \(\displaystyle{x}={a}{y}^{{2}}+{b}{y}+{c}\) .

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asked 2021-08-11

\(\displaystyle{S}_{{1}},{S}_{{2}}\) are two foci of the ellipse \(\displaystyle{x}^{{2}}+{2}{y}^{{2}}={2}\). P be any point on the ellipse The locus incentre of the triangle \(\displaystyle{P}{S}{S}_{{1}}\), is a conic where length of its latus rectum is

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asked 2021-08-11

Find the vertices and foci of the conic section. \(\frac{x^2}{4 }− \frac{y^2}{9} = 36\)

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asked 2021-08-11

Find the equation of the graph for each conic in standard form. Identify the conic, the center, the vertex, the co-vertex, the focus (foci), major axis, minor axis, \(\displaystyle{a}^{{2}},{b}^{{2}}\), and \(\displaystyle{c}^{{2}}\)?. For hyperbola, find the asymptotes. Sketch the graph. \(\displaystyle{9}{x}^{{2}}+{25}{y}^{{2}}+{54}{x}-{50}{y}-{119}={0}\)

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asked 2021-08-11

Find the equation of the graph for each conic in general form. Identify the conic, the center, the vertex, the co-vertex, the focus (foci), major axis, minor axis, \(\displaystyle{a}^{{2}}\), \(\displaystyle{b}^{{2}}\), and \(\displaystyle{c}^{{2}}\). For hyperbola,find the asymtotes.Sketch the graph
\(\displaystyle{16}{x}^{{2}}-{4}{y}^{{2}}-{64}={0}\)

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asked 2021-08-10

Show that the graph of an equation of the form
\(\displaystyle{A}{x}^{{2}}+{C}{y}^{{2}}+{D}{x}+{E}{y}+{F}={0},{A}\ne{0},{C}\ne{0}\)
where A and C are of opposite sign,
(a) is a hyperbola if \(\displaystyle{\frac{{{D}^{{2}}}}{{{4}{A}}}}+{\frac{{{E}^{{2}}}}{{{4}{C}}}}-{F}\ne{0}\)
(b) is two intersecting lines if \(\displaystyle{\frac{{{D}^{{2}}}}{{{4}{A}}}}+{\frac{{{E}^{{2}}}}{{{4}{C}}}}-{F}={0}\)

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asked 2021-08-10

Let A be a symmetric \(\displaystyle{2}\times{2}\) matrix and let k be a scalar. Prove that the graph of the quadratic equation \(\displaystyle{x}^{{T}}\) Ax=k is ,an ellipse, circle, or imaginary conic if \(\displaystyle{k}\ne{q}{0}\) and det A > 0

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asked 2021-08-10

Perform a rotation of axes to eliminate the xy-term. xy+5=0

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asked 2021-08-10

Find the equation of the graph for each conic in general form. Identify the conic, the center, the vertex, the co-vertex, the focus (foci), major axis, minor axis, \(\displaystyle{a}^{{2}}\), \(\displaystyle{b}^{{2}}\), and \(\displaystyle{c}^{{2}}\). For hyperbola,find the asymtotes.Sketch the graph
\(\displaystyle{2}{\left({y}+{2}\right)}^{{2}}+{4}{\left({x}-{4}\right)}^{{2}}={8}\)

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