RECTANGULAR EQUATIONS OF CONICS A quick overview of the 4 conic sections in rectangular coordinates is presented below. 1. Circles Skipped covered in MAT 124 (Precalculus I). 2. s Definition A parabola is the set of all points in the plane such that, where is a fixed point in the plane and is a fixed line in the plane. These are respectively called the focus and the directrix of the parabola. A parabola is then the set of all the points that are equidistant to both the focus and the directrix. is the vertex of the parabola. This is the point where the parabola intersects its axis of symmetry (See Fig. 3 on p. 655). Note that the vertex is the point on the parabola that is closest to the directrix. Here we let. If, these are the four cases: Focus Equation of Directrix Equation of Axis of Symmetry Description of x-axis Opens right x-axis Opens left y-axis Opens up y-axis Opens down
If, these are the four cases: Focus Equation of Directrix Equation of Axis of Symmetry Description of Parallel to x-axis Opens right Parallel to x-axis Opens left Parallel to y-axis Opens up Parallel to y-axis Opens down Example Analyze the equation. Answer Here we have Thus we infer the following: This describes a parabola that opens up with a vertex at. Note that to find the two x-intercepts of this parabola, it suffices to plug-in in the equation of the parabola. This yields, which implies that, or. So the two x-intercepts are the points and. [Graph the parabola to confirm all these analytical results.]
3. Ellipses Definition An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points and, called the foci, is a constant. Here we let, where. is the center of the ellipse, and, are the two vertices of the ellipse. The vertices are the points where the ellipse intersects its major axis. Note that,,, and are all points on the major axis. Here we let,, and denote the distance between the center of the ellipse and the two points where the ellipse intersects its minor axis (See Figs. 19 & 20 on p. 665). If and the major axis is along the x-axis, we then have the following: Equation of ellipse:, where and (See Fig. 21 on p. 666) If and the major axis is along the y-axis, we then have the following: Equation of ellipse:, where and (See Fig. 25 on p. 668) If and the major axis is parallel to either the x-axis or the y-axis, then replace with (horizontal shift) and with (vertical shift) in the previous equations for ellipses. This is all summarized in Table 3 on p. 670.
Example Analyze the equation. Answer Here we have Thus we infer that this is an ellipse with center whose major axis is parallel to the y- axis (it s the vertical line ) and whose minor axis is parallel to the x-axis (it s the horizontal line ). We also have the following: Therefore the vertices of the ellipse are the points and, and its foci are the points and. Note that since and, the two points where the ellipse intersects its minor axis are given by. [Check this with the equation of the ellipse.] [Graph the ellipse to confirm all these analytical results.] 4. Hyperbolas Definition A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points and, called the foci, is a constant. Here we let, where., the center of the hyperbola, is the point where the transverse axis and the conjugate axis of the hyperbola meet at a right angle. and, the two vertices of the hyperbola, are the points where the hyperbola intersects its transverse axis. Note that,,, and are all points on the transverse axis.
Here we let and. Note that a hyperbola consists of two separate branches (See Figs. 34 & 35 on p. 677) and has two oblique asymptotes (See Fig. 43 on p. 682). If and the transverse axis is along the x-axis, we then have the following: Equation of hyperbola:, where (See Fig. 36 on p. 678) Its two oblique asymptotes are the lines and. If and the transverse axis is along the y-axis, we then have the following: Equation of hyperbola:, where (See Fig. 40 on p. 680) Its two oblique asymptotes are the lines and. If and the transverse axis is parallel to either the x-axis or the y-axis, then replace with (horizontal shift) and with (vertical shift) in the previous equations for hyperbolas. This is all summarized in Table 4 on p. 683.
Example Analyze the equation. Answer Here we have Thus we infer that this is a hyperbola with center whose transverse axis is parallel to the y-axis (it s the vertical line ) and whose conjugate axis is parallel to the x-axis (it s the horizontal line ). We also have the following: Therefore the vertices of the hyperbola are the points and, and its foci are the points and. Moreover, the hyperbola has two oblique asymptotes: and, or and. [Graph the hyperbola to confirm all these analytical results.]