ADVANCED EXERCISES - Circles and other Conic Sections
(1) Find the equations for the parabolas described below:
(a) vertex is 2 units from the x-axis, opens downward, symmetric about x = 1, y-intercept = 23/12
(b) has vertical axis of symmetry and passing through (-2,3), (0,3), and (1,9).
(c) Opens upward, passing through (-2,7), vertex is on the positive y-axis and is 5 units from the origin.
(d) Find an equation that represents a parabola which opens to the right, has its vertex at (2,2) and passes through the point (5,0).
(2) Given a circle with radius 1 unit and center at (0,0). A rectangle is inscribed inside the circle such that one of its sides is parallel to the x-axis and crosses the y-axis at y. Express the folowing as functions of x.
(a) The perimeter of the rectangle
(b) The area of the rectangle
(3) In geometry, it is shown that three points in the plane, not in a straight line, determine a unique circle which passes through those three points.
(a) Find the equation of the circle which passes through (0,0), (0,1) and (2,0).
(b) Describe a procedure that will always work to find the circle which passes through three given noncollinear points.
(a) Sketch the graph of y = 1/x or, equivalently, xy = 1
(b) What shape does the graph drawn in (a) look like?
(5) From the following graphs, determine the equation that formed them:
(a)
Vertex (3, 2) Focus (3, 3)
(b)
Center (-4, -2), P (4, -2)
(c)
Center (3, -3), P (3, 1).
