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Chapter 3. Conic Sections. Section 3.2. Translation of Axes. Vertical Translation.
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Chapter 3 Conic Sections
Section 3.2 Translation of Axes
Vertical Translation • If all y in the equation y = E is replaced by (y – C), where E is an expression involving x and C is a real number, then the graph of the new equation is the same as the graph of y = E, but translated C units upward if C > 0, or |C| units downwards if C < 0.
Horizontal Translation • If all x in the equation y = E is replaced by (x – C), then the graph of the new equation y = E’ is the same as the graph of y = E, but translated C units to the right if C > 0 and |C| units to the left if C < 0.
Example 1 • Determine how many units and in what direction the graph of the first equation must be shifted to obtain the second equation. (a) x2 + y2 = 12; (x – 2)2 + (y + 3)2 = 12 (b) x2 = 2y; (x+5)2 = 2y – 14 (c)
Example 2 • Give the resulting equation if the graphs of the given equations are to be shifted 2 units up and 3 units to the left: (a) x2 + y2 = 100 (b) 4x2 + y2 + 12x – 9y = 0
Sections 3.1, 3.3 Introduction to Conic Sections; Parabolas
Definition • A parabola is the set of all points in a plane equidistant from a fixed point F and a fixed line L not containing F. • The fixed point is called the focus of the parabola and the fixed line the directrix.
Definition • The line through the focus perpendicular to the directrix is called the axis of symmetry of the parabola. • The vertex is the single point on the parabola that is on the axis of symmetry. • The distance from the vertex to the focus (or the directrix) is called the focal distance, which shall be denoted by p, where p > 0.
Remarks • Keep in mind that the directrix is of an equal distance to the vertex as the focus of the parabola, but in the opposite side. • Expanding the equation of a parabola, we see that the general form of a parabola is Ax2 + Cy2 + Dx + Ey + F = 0 where exactly one of A and C is zero.
Example 1 • Write an equation of the parabola satisfying the following properties: (a) Vertex at (6,8), directrix at x = 1. (b) Directrix at y = 2 and focus at (2,-4). (c) Focus (-1,3), passing through (3,6), axis parallel to the y-axis.
Graphs of Parabolas • To sketch the graph of a parabola if its equation is known, one needs three points: the vertex, and one point from each “branch” of the parabola.
Example 2 • For the parabolas below, find the focal distance, the coordinates of the vertex and the focus, as well as the equations of the axis of symmetry and the directrix. Then sketch their graphs. (a) x2 + 2x – y + 3 = 0 (b) y2 – 8y + 6x = 0
Example 3 • Find an equation of the parabola whose axis is horizontal, and passes through (6,2), (-3,-1), and (-2,0).
Definition • The latus rectum of a parabola is the line segment joining two points of the parabola, perpendicular to the axis of symmetry, and passing through the focus. • It can be shown that the length of the latus rectum is 4a, where a is the focal distance.
Example 4 • The endpoints of the latus rectum of a parabola are (4,-k) and (4,k), where k>0. If the directrix of the parabola passes through the point (-2,10), find k and the equation of the parabola.
Example 5 • A waterway for a boat ride in a recreational park has a parabolic cross-section. This waterway has a depth of 12 feet and a width of 6 feet at the surface. Boats with rectangular cross-section will be used and they are found to sink to a depth of 4 feet when placed on the water-filled waterway. What is the maximum width that these boats can have so that they will float and move smoothly on the waterway?
Section 3.4 Ellipses
Definition • An ellipse is the set of all points in a plane the sum of whose distances from two fixed points F1 and F2 is a constant. Each of the two fixed points is called a focus of the ellipse.
Definition • The line through the foci of an ellipse is called its principal axis. • The two points of the ellipse that lie on the principal axis are called vertices and the line segment joining them is called the major axis. • The midpoint of the major axis is called the center of the ellipse.
Definition • The distance from the center to a focus is called the focal distance. • The line segment through the center of an ellipse, perpendicular to the major axis and whose endpoints are on the ellipse is called the minor axis. Its endpoints are called co-vertices.
Remark • We shall use the following symbols: • a = distance from the center to a vertex • b = distance from the center to a co-vertex • c = the focal distance
Some Properties of an Ellipse • The sum of the distances from any point of an ellipse to the two foci is 2a, where a is the distance from the center to the vertex. • The constants a, b, c are related by the equation a2 = b2 + c2. • This means that the major axis of an ellipse is always longer than its minor axis.
Remarks • Keep in mind that one can spot the orientation of the major axis of an ellipse given its equation in standard form by examining which variable has a larger denominator. • The center of an ellipse is equidistant from the foci. • Expanding the equation of an ellipse, we see that the general form of an ellipse is Ax2 + Cy2 + Dx + Ey + F = 0 where AC > 0.
Example 1 • Write an equation of the ellipse(s) satisfying the following properties: (a) Center (0,0), length of minor axis 10, distance between foci 24. (b) One vertex at (3,2), one co-vertex at (7,1). (c) Center (1,1), a vertex at (1,3), and which passes through the origin.
Graphs of Ellipses • To sketch the graph of an ellipse where the equation is given, we begin by plotting the center of the ellipse. • We then count a units (to both sides) from the center along the major axis to plot the vertices. The co-vertices are obtained by counting b units (to both sides) from the center along the minor axis.
Example 2 • For the ellipses below, find the coordinates of the center, the vertices, the foci, and the co-vertices. Then sketch their graphs. (a) 36(x+3)2 + 4y2 – 144 = 0 (b) 4x2 + 8y2 + 4x + 24y – 13 = 0
Ellipse, point, or null set? • Not all equations of the form Ax2+Cy2+Dx+Ey+F=0 where A, C > 0 are those of ellipses. To determine whether the equation is that of an ellipse, we write the equation in the form A(x-h)2 + C(y–k)2 = a. • If a>0, then the graph is an ellipse. • If a=0, then the graph is a single point (h, k). • If a<0, then the graph is a null set.
Example 3 • Determine whether the graph of each of the following equations is an ellipse, a point, or an empty set: (a) x2 + 36y2 + 6x – 72y + 45 = 0 (b) 9x2 + 16y2 + 90x – 160y + 481 = 0
Example 4 • A one-way road has an overpass in the form of a semi-ellipse, 15 feet high at the center, 40 feet wide. Assuming a truck is 12 feet wide, what is the maximum height of such a truck that can pass under the overpass?
Section 3.5 Hyperbolas
Definition • A hyperbola is the set of all points in the plane the absolute value of the difference of whose distances from two distinct fixed points F1 and F2 is a constant. These two fixed points are the foci of the hyperbola.
Definition • The line through the foci is called the principal axis. • The two points of the hyperbola that lie on the principal axis are called vertices, and the line segment joining them is the transverse axis. • The midpoint of the transverse axis is the center of the hyperbola.
Remark • We shall be using the following symbols: • a = distance from the center to a vertex • c = the focal distance • = distance from the center to one endpoint of the conjugate axis, which is the line segment with length 2b, perpendicular to the transverse axis and whose midpoint is the center of the hyperbola.
Some Properties of Hyperbola • The absolute value of the difference of the distances of any point of a hyperbola from the two foci is 2a, where a is the distance from the center to a vertex. • The conjugate axis of a hyperbola may be shorter or longer than the transverse axis. • From the definition of the conjugate axis, we know that c2 = a2 + b2.
Remarks • A hyperbola is composed of two non-intersecting curves called its branches. These branches approach two diagonal lines called asymptotes. • For a hyperbola with center (h,k), the equations of the asymptotes are: if the transverse axis is horizontal if the transverse axis is vertical
Remarks • Expanding the equation of a hyperbola, we see that the general form of a hyperbola is Ax2 + Cy2 + Dx + Ey + F = 0 where AC < 0.
Example 1 • Write an equation of the hyperbola satisfying the following properties: (a) Center at (0,0), transverse axis along the y-axis, passing through points (5,3) and (-3,2). (b) Foci at (-1,4) and (7,4), transverse axis of length 8/3. (c) Asymptotes are y – x – 1 = 0 and y + x – 1 = 0; y-intercepts are 3 and -1.
Graphs of Hyperbolas • To sketch the graph of a hyperbola where the equation is given, we need to construct an auxiliary rectangle. • From the center of the hyperbola, count a units from the center to both sides of the center. Then in the direction perpendicular to the transverse axis, count b units from the center to both sides of the center. • Draw a rectangle whose sides have these four points as midpoints. Joining the opposite vertices of the rectangle yields the asymptotes of the hyperbola.
Example 2 • For the hyperbolas below, find the coordinates of the center, the vertices, the foci, and the equations of the asymptotes. Then sketch their graphs. (a) 9x2 – 4y2 – 36x – 16y – 16 = 0 (b) 49y2 – 4x2 – 98y + 48x – 291 = 0
Hyperbolas or intersecting lines? • Not all equations of the form Ax2+Cy2+Dx+Ey+F=0 where AC < 0 are those of hyperbolas. To determine whether the equation is that of a hyperbola or just two intersecting lines, we write the equation in the form A(x–h)2 – C(y–k)2 = a (or A(y–k)2 – C(x–h)2 = a). • If a0, then the graph is a hyperbola. • If a=0, then the graph is a pair of intersecting lines.
