Parabolas and Circles

PARABOLAS AND CIRCLES

Parabolas

·  A parabola is the set of all ______from a line and a ______not on the line.

·  The line is called the ______, and the point is called the ______.

·  The point on the parabola halfway between the focus and the directrix is the ______.

·  The line containing the focus and the vertex is the ______.

·  A parabola is symmetric with respect to its ______.

·  The ______is the chord parallel to the directrix and passing through the ______.

Vertical Parabolas Horizontal Parabolas

Form à Form à

Vertex à Vertex à

Axis of Symmetry à Axis of Symmetry à

Focus à Focus à

Directrix à Directrix à

Opening à Opening à

Length of latus rectum à Length of latus rectum à

1)  If either A = 0 or B = 0 then the equation defines a ______(x² or y² is missing).

2)  Isolate the ______that is not squared and use the completing the square method to convert the equation to that of a Parabola in ______.

3)  Solve for ______if there is no y² in the equation or solve for ______if there is no x² in the equation.

Example: Find the vertex, focus, axis of symmetry, direction of the opening, directrix, and length of the latus rectum of the following parabola: x2 - 4x - 8y + 28 = 0. Then graph the parabola. (Hint: First put the equation in standard form.)

Example: Write the equation of a parabola with a focus of (3, 5) and a directrix of y = 1.

Circles:

·  A ______is the collection of points equidistant from a fixed point.

·  The fixed point is called the ______.

·  The distance from the center to any point on the circle is the ______of the circle, and a segment containing the center whose endpoints are both on the circle is a ______of the circle.

·  The radius, r, equals ______the diameter, d.

The standard equation for a circle is ______.

The center is at ______

The radius is ______

Example: Write the equation of a circle with center (-4,3) and radius 6.

Example: Write the equation of in (h, k) form. Find the center, radius, domain, and range of the circle. Sketch the circle.

Normal and Tangent Lines to a Circle

A line is said to be ______ to a curve at a point if it is perpendicular to the ______ to the curve at that point.

Example: Given, find equations of the normal and tangent lines at point P (5, -12).