双曲线

In analytic geometry a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. As with the ellipse, every hyperbola has two axes of symmetry . The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. Note that the vertices, co-vertices, and foci are related by the equation c2 =a2+b2. When we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci. How To: Given the equation of a hyperbola in standard form, locate its vertices and foci. 1. Determine whether the transverse axis lies on the x – or y -axis. Notice that a2 is always under the variable with the positive coefficient. So, if you set the other variable equal to zero, you can easily find the intercepts. In the case where the hyperbola is centered at the origin, the intercepts coincide with the vertices. Identify the vertices and foci of the hyperbola with equation y249−x232 =1. Identify the vertices and foci of the hyperbola with equation x29−y225 =1. Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin. Reviewing the standard forms given for hyperbolas centered at (0,0), we see that the vertices, co-vertices, and foci are related by the equation c2 =a2+b2. Note that this equation can also be rewritten as b2 =c2−a2. This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices. What is the standard form equation of the hyperbola that has vertices (±6,0) and foci (±210,0)? The vertices and foci are on the x -axis. Thus, the equation for the hyperbola will have the form x2a2−y2b2 =1. What is the standard form equation of the hyperbola that has vertices (0,±2) and foci (0,±25)? Using the reasoning above, the equations of the asymptotes are y =±ab(x−h)+k. Like hyperbolas centered at the origin, hyperbolas centered at a point (h,k) have vertices, co-vertices, and foci that are related by the equation c2 =a2+b2. We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given. The y -coordinates of the vertices and foci are the same, so the transverse axis is parallel to the x -axis. Thus, the equation of the hyperbola will have the form Finally, substitute the values found for h,k,a2, and b2 into the standard form of the equation. What is the standard form equation of the hyperbola that has vertices (1,−2) and (1,8) and foci (1,−10) and (1,16)? As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy,