9.1 Circles And Parabolas
A conic section or conic is the intersection of a plane and a double napped cone.
Conics can be defined algebraically, in terms of the general second degree equationAx² + Bxy + Cy² + Dx + Ey + F = 0
they could also be defined through a locus, a collection of points satisfying certain geometric property.
they could also be defined through a locus, a collection of points satisfying certain geometric property.
Circles
A circle is the set of all points (x, y) in a plane that are equidistant from a fixed point (h, k), called the center of the circle. The distance r between the center and any point (x, y) on the circle is the radius. The standard form of the equation of a circle is (x - h)² + (y - k)² = r² The point (h, k) is the center of the circle, and the positive number r is the radius of the circle. The standard form of the equation of a circle whose center is the origin, (h, k) = (0, 0), is x² + y² = r² |
Finding the Standard Equation of a Circle
Sketching a Circle
Finding the Intercepts of a Circle
Parabolas
A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line. The midpoint between the focus and the directrix is the vertex, and the line passing through the focus and the vertex is the axis of the parabola. The standard form of the equation of a parabola with vertex at (h, k) (x - h)² = 4p(y - k), p ≠ 0 vertical axis, directrix y = k - p (y - k)² = 4p(x - h), p ≠ 0 horizontal axis, directrix x = h - p The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0, 0), then the equation takes one of the following forms. x² = 4py y² = 4px |
Finding the Standard Equation of a Parabola
Finding the Focus of a Parabola
A line is tangent to a parabola at a point on the parabola when the line intersects, but does not cross, the parabola at the point. Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces.
The tangent line to a parabola at a point P makes equal angles with the line passing through P and the focus, and the axis of the parabola. |
Finding the Tangent Line at a Point on a Parabola