# Instructions: Graph the conic section and make sure to label the coordinates in the graph. Include all the calculations needed to complete the graph. Give the standard form (SF) and the general form (GF) of the conic sections. HYPERBOLA: 1) The vertices are at (-2, 0) and (2, 0). The conjugate axis' length is 6.

Question
Conic sections
Instructions: Graph the conic section and make sure to label the coordinates in the graph. Include all the calculations needed to complete the graph. Give the standard form (SF) and the general form (GF) of the conic sections. HYPERBOLA: 1) The vertices are at (-2, 0) and (2, 0). The conjugate axis' length is 6.

2020-12-04
Given: The vertices are at (-2, 0) and (2, 0). The conjugate axis's length is 6. The two vertices have the same y coordinate, therefore, the hyperbola is the horizontal transverse axis type. The standard equation for the horizontal transverse axis type is $$\frac{(x - h)^2}{a^2} - ((y - k)^2)/(b^2) = 1$$ We know that the general form for the vertices of this type is: $$(h + a, k) and (h - a, k)$$ comparing with vertices $$k = 0$$
$$h + a = 2$$
$$h - a = -2$$ Solve $$h = 0, a = 2$$ The length of the conjugate axis is equal to 2b $$2b = 6$$
$$b = 3$$ Standard form $$\frac{(x - 0)^2}{2^2} + \frac{(y - 0^2)}{3^2} = 1$$
$$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ General form $$9x^2 + 4y^2 = 36$$
$$9x^2 + 4y^2 - 36 = 0$$

### Relevant Questions

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 $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$
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)
Find the equation of the graph for each conic in standard form. Identify the conic, the center, the co-vertex, the focus (foci), major axis, minor axis, $$a^{2}, b^{2},\ and c^{2}.$$ For hyperbola, find the asymptotes $$9x^{2}\ -\ 4y^{2}\ +\ 54x\ +\ 32y\ +\ 119 = 0$$
Solve, a. If necessary, write the equation in one of the standard forms for a conic in polar coordinates $$r = \frac{6}{2 + sin \theta}$$ b. Determine values for e and p. Use the value of e to identify the conic section. c. Graph the given polar equation.
Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: $$x = h + r cos(?), y = k + r sin(?)$$ Use your result to find a set of parametric equations for the line or conic section. $$(When 0 \leq ? \leq 2?.)$$ Circle: center: (6, 3), radius: 7
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. $$x2\+\ y2\ −\ 4x\ −\ 6y\ +\ 1 = 0$$
Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e = 2,\ x = 4$$
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.