If the directrix is horizontal, then the parabola opens (text{horizontally}/text{vertically})

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asked 2021-03-05

A latus rectum of a conic section is a chord through a focus parallel to the directrix. Find the area bounded by the parabola \(\displaystyle{y}={x}^{2}\text{/}{\left({4}{c}\right)}\) and its latus rectum.

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asked 2021-01-10

Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3,3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2,4) and focus at (-4, 4)

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asked 2021-01-06

Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3, 3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2, 4) and focus at (-4, 4)

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asked 2020-12-24

For Exercise, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is. If the equation represents a circle, identify the center and radius. • If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. • If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. • If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. \(x2\+\ y2\ −\ 4x\ −\ 6y\ +\ 1 = 0\)

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asked 2020-11-11

If a conic section is written as a polar equation, and the denominator involves \(\displaystyle{\sin{\ }}\theta\), what conclusion can be drawn about the directrix?

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asked 2020-11-27

We need to verify theorem 5 Focus-Directrix definition: \(\displaystyle{0}\ {<}\ {e}\ {<}\ {1}\ \text{and}\ {c}={d}{\left({e}\ -\ {2}\ -\ {1}\right)}\)

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asked 2020-11-24

Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.
\(\displaystyle{\left({a}\right)}{4}{x}{2}-{9}{y}{2}={12}{\left({b}\right)}-{4}{x}+{9}{y}{2}={0}\)
\(\displaystyle{\left({c}\right)}{4}{y}{2}+{9}{x}{2}={12}{\left({d}\right)}{4}{x}{3}+{9}{y}{3}={12}\)

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asked 2021-03-11

An automobile tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles). A mathematical model for the data is given by
\(\displaystyle f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.\)
\(\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
(A) Complete the table below.
\(\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
(Round to one decimal place as needed.)
\(A. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
\(B. 20602060xf(x)\)
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
\(C. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
\(D.20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
(C) Use the modeling function f(x) to estimate the mileage for a tire pressure of 29
\(\displaystyle\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) and for 35
\(\displaystyle\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\)
The mileage for the tire pressure \(\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) is
The mileage for the tire pressure \(\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}\) in. is
(Round to two decimal places as needed.)
(D) Write a brief description of the relationship between tire pressure and mileage.
A. As tire pressure increases, mileage decreases to a minimum at a certain tire pressure, then begins to increase.
B. As tire pressure increases, mileage decreases.
C. As tire pressure increases, mileage increases to a maximum at a certain tire pressure, then begins to decrease.
D. As tire pressure increases, mileage increases.

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asked 2020-10-25

Consider the equation:
\(\displaystyle{2}\sqrt{{3}}{x}^{2}-{6}{x}{y}+\sqrt{{3}}{x}+{3}{y}={0}\)
a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola?
b) Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies \(\displaystyle{0}\le\varphi\le\pi\text{/}{2}\))

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asked 2021-02-15

Determine whether the statement If \(\displaystyle{D}\ne{0}{\quad\text{or}\quad}{E}\ne{0}\),
then the graph of \(\displaystyle{y}^{2}-{x}^{2}+{D}{x}+{E}{y}={0}\) is a hyperbolais true or false. If it is false, explain why or give an example that shows it is false.

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Answer 1

Step 1
Given:
We must find the missing word in the given sentence.
So, Definition of parabola said:
A parabola is another type of conic section generated by the cross section of a cone intersected by a plane.
An equation of the form:
\(\displaystyle{y}={a}{x}^{2}+{b}{x}+{c}{\left({a}\ne{0}\right)}\)
Step 2
The above equation is parabola opening upward if \(\displaystyle{a}>{0}\) and opening downward if \(\displaystyle{a}<{0}.\)</span>
So, the geometrically definition of parabola is the set of all points in a plane that the equidistant from a fixed line called directrix and a fixed point called the focus.
Step 3
In the standard form of a parabola, if x and y are replaced by \(\displaystyle{x}-{h}{\quad\text{and}\quad}{y}-{k}\), then the graph of the parabola is shifted h units horizontally and k units vertically. The vertex is (h, k), and the focus is \(\displaystyle{\left({h},{k}+{p}\right)}.\)
The directrix is the line defined by the equation \(\displaystyle{y}={k}-{p}\) and the equation of the parabola is:
\(\displaystyle{\left({x}-{h}\right)}^{2}={4}{p}{\left({y}-{k}\right)}\)
Step 4
The equation of directrix \(\displaystyle{y}={k}-{p}\) is horizontal. So, the parabola opens towards vertically.
Thus, the missing word is Vertically.

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If the directrix is horizontal, then the parabola opens (text{horizontally}/text{vertically})

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