INTRODUCTORY EXERCISES - Circles and other Conic Sections
(1) By completing the square, show that the following equations represent a circle and find its radius and center.
(a) x2 + y2 + 2x - 6y + 7 = 0
(b) x2 + y2 - 4x + 10y + 13 = 0
(c) 16x2 + 16y2 + 8x + 32y + 1 = 0
(2) Find the equation of each circle.
(a) radius 5 and center at (-1,9)
(b) radius 10 and center at (-2,-8)
(3) Write the equation for the parabolas.
(a) vertex (0,0), passing through (1,12) and symmetric about the axis x = 0
(b) vertex (2,0), symmetric about the x-axis and passing through (3,2)
(4) Find an equation that represents an ellipse centered at the origin whose major axis is 12 units long and whose minor axis is 2 units long. Sketch this ellipse.
(5) Sketch the following regions that are bounded by each pair of curves (find the points of intersection first):
(a) x-axis and the parabola y = x2 - 1.
(b) circle x2 + y2 = 4, the line x = 1 and the y-axis.
(c) the parabola y = x2 and the 45o line from quadrant I to quadrant III.
(d) the parabola x = -y2 + 1 and the line y = -x.
