# Solve for the equation in standard form of the following conic sections and graph the curve on a Cartesian plane indicating important points. 1. An ellipse passing through (4, 3) and (6, 2) 2. A parabola with axis parallel to the x-axis and passing through (5, 4), (11, -2) and (21, -4) 3. The hyperbola given by displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.

Question
Conic sections
Solve for the equation in standard form of the following conic sections and graph the curve on a Cartesian plane indicating important points.
1. An ellipse passing through (4, 3) and (6, 2)
2. A parabola with axis parallel to the x-axis and passing through (5, 4), (11, -2) and (21, -4)
3. The hyperbola given by $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$

2020-11-11
Step 1
An equation of the ellipse is of the form,
$$\displaystyle\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}={1}$$
Equation of the ellipse through the point (4, 3).
So, $$\displaystyle{x}={4},{y}={3}$$ satisfies the equation of an ellipse.
Implies that,
$$\displaystyle\frac{{4}^{2}}{{a}^{2}}+\frac{{3}^{2}}{{b}^{2}}={1}$$
$$\displaystyle\Rightarrow\frac{16}{{a}^{2}}+\frac{9}{{b}^{2}}={1}$$
$$\displaystyle\Rightarrow\frac{16}{{a}^{2}}={1}-\frac{9}{{b}^{2}}$$
$$\displaystyle\Rightarrow\frac{1}{{a}^{2}}=\frac{1}{{16}}{\left({1}-\frac{9}{{b}^{2}}\right)}$$
Step 2
Equation of the ellipse through the point (6, 2).
So, $$\displaystyle{x}={6},{y}={2}$$ satisfies the equation of an ellipse.
$$\displaystyle\Rightarrow\frac{{6}^{2}}{{a}^{2}}+\frac{{2}^{2}}{{b}^{2}}={1}$$
$$\displaystyle\Rightarrow\frac{36}{{a}^{2}}+\frac{4}{{b}^{2}}={1}$$
$$\displaystyle\Rightarrow\frac{36}{{a}^{2}}={1}-\frac{4}{{b}^{2}}$$
$$\displaystyle\Rightarrow\frac{1}{{a}^{2}}=\frac{1}{{36}}{\left({1}-\frac{4}{{b}^{2}}\right)}$$
Step 3
Compare equations (1) and (2), we get
$$\displaystyle\frac{1}{{36}}{\left({1}-\frac{4}{{b}^{2}}\right)}=\frac{1}{{16}}{\left({1}-\frac{9}{{b}^{2}}\right)}$$
Multiply both sides by 4,
$$\displaystyle\frac{1}{{9}}{\left({1}-\frac{4}{{b}^{2}}\right)}=\frac{1}{{4}}{\left({1}-\frac{9}{{b}^{2}}\right)}$$
Distribute both sides,
$$\displaystyle\frac{1}{{9}}-\frac{4}{{{9}{b}^{2}}}=\frac{1}{{4}}-\frac{9}{{{4}{b}^{2}}}$$
Combine like terms
$$\displaystyle\frac{1}{{9}}-\frac{1}{{4}}=\frac{4}{{{9}{b}^{2}}}-\frac{9}{{{4}{b}^{2}}}$$
Make the same denominator,
$$\displaystyle\frac{4}{{36}}-\frac{9}{{36}}=\frac{16}{{{36}{b}^{2}}}-\frac{81}{{{36}{b}^{2}}}$$
$$\displaystyle\Rightarrow-\frac{5}{{36}}=-\frac{65}{{{36}{b}^{2}}}$$
$$\displaystyle\Rightarrow-\frac{5}{{36}}=-\frac{65}{{{36}{b}^{2}}}$$
Multiply both sides by 36,
$$\displaystyle\Rightarrow-\frac{5}{{1}}=-\frac{65}{{b}^{2}}$$
Divide both sides by -65,
$$\displaystyle\Rightarrow\frac{5}{{65}}=\frac{1}{{b}^{2}}$$
By simplifying it,
$$\displaystyle\Rightarrow\frac{1}{{13}}=\frac{1}{{b}^{2}}$$
Taking reciprocal from both sides,
$$\displaystyle\Rightarrow{b}^{2}={13}$$
Step 4
Substitute $$\displaystyle{b}^{2}={13}$$
in the equation $$\displaystyle\frac{1}{{a}^{2}}=\frac{1}{{16}}{\left({1}-\frac{9}{{b}^{2}}\right)},$$
$$\displaystyle\Rightarrow\frac{1}{{a}^{2}}=\frac{1}{{16}}{\left({1}-\frac{9}{{13}}\right)}$$
$$\displaystyle\Rightarrow\frac{1}{{a}^{2}}=\frac{1}{{16}}{\left(\frac{4}{{13}}\right)}$$
$$\displaystyle\Rightarrow\frac{1}{{a}^{2}}=\frac{1}{{4}}{\left(\frac{1}{{13}}\right)}$$
Taking reciprocal from both sides
$$\displaystyle\Rightarrow{a}^{2}={52}$$
The equation of ellipse through the points (4, 3) and (6, 2) is $$\displaystyle\frac{{x}^{2}}{{52}}+\frac{{y}^{2}}{{13}}={1}$$.

### Relevant Questions

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)
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)
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 $$\displaystyle{0}\le\varphi\le\pi\text{/}{2}$$)
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.
Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.
$$\displaystyle{\left({a}\right)}{4}{x}{2}-{9}{y}{2}={12}{\left({b}\right)}-{4}{x}+{9}{y}{2}={0}$$
$$\displaystyle{\left({c}\right)}{4}{y}{2}+{9}{x}{2}={12}{\left({d}\right)}{4}{x}{3}+{9}{y}{3}={12}$$
$$\displaystyle{9}{x}{2}+{4}{y}{2}-{24}{y}+{36}={0}$$
The circle, ellipse, hyperbola, and parabola are examples of conic sections. Their quation contains $$x^2 terms, y^2$$ terms, or both. When these terms both appear, are on the same side, and have different coefficients with same signs, the equation is that of an ellipse.
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.
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. $$x2\+\ y2\ −\ 4x\ −\ 6y\ +\ 1 = 0$$