Tiêu đề 63 Conic Sections - Parabolas.pdf ✅

MSTC 62: Conic Sections – Parabolas 1. Locus A parabola is the locus of points (x, y) equidistant to a point called the focus and a line called the directrix. For parabolas with a vertical or horizontal axis of symmetry, the standard equation typically is a quadratic equation. 2. Equations Consider a quadratic equation, y = Ax 2 + Bx + C By completing the square, y − C = A (x 2 + B A x) y − C + B 2 4A = A (x 2 + B A x + B 2 4A2 ) y − C + B 2 4A = A (x + B 2A ) 2 1 A (y − C + B 2 4A ) = (x + B 2A ) 2 Let k = C − B 2 4A , h = − B 2A , 4a = 1 A , 4a(y − k) = (x − h) 2 This equation is called the vertex form of the equation of vertical parabolas. Similarly, if the equation is quadratic in y, 4a(x − h) = (y − k) 2 3. Properties For parabolas expressed in their vertex form, Characteristic Definition Value Vertex The corner point at which the parabola makes its maximum turn. (h, k) Focal Distance The distance from the vertex to the focus. The distance between the foci and directrix is 2a. |a| Direction The direction in which the parabola opens towards. If a is positive, it opens upward or to the right based on the orientation. If a is negative, it opens downward or to the left based on the orientation. Eccentricity The ratio of the distance of the focal distance to the vertex distance. The eccentricity of a parabola is always 1. e = 1 Directrix A line parallel to the latus rectum of the parabola is perpendicular to its major axis. x = h − a if horizontal y = k − a if vertical Latus Rectum A segment that passes through the foci and is perpendicular to the major axis, which intersects the conic. LR = |4a|