Pre-Calculus. 2) Find the equation of the circle having (2, 5) and (-2, -1) as endpoints of the diameter.

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1 Pre-Calculus Conic Review Name Block Date Circles: 1) Determine the center and radius of each circle. a) ( x 5) + ( y + 6) = 11 b) x y x y = 0 ) Find the equation of the circle having (, 5) and (-, -1) as endpoints of the diameter. 3) Write the equations for the following circles. a) Center (6, 1); radius of 5 b) Center (4, -) and passes through (6, 3) Ellipses: 4) Determine the vertices, minor axis end points, and foci of each ellipse. x y a) + = 1 b) 5x + 16y = ) Find the equation of the following ellipses. a) 5x + y 50x + 1y 39 = 0 b) foci (, 7) and (-6, 7), major axis of length 10 Parabolas: 6) Find the coordinates of the vertex, focus and the equation of the directrix for the following parabolas. 1 a) y = x b) ( y + 1) = 4( x 3) 6

2 7) Find an equation for the parabola whose directrix is x = -3 and whose focus is (3, 0). 8) Determine an equation in standard form for each parabola. a) focus (, 5); directrix: y = -1 b) vertex (-4, 3); directrix: x = 1 Hyperbolas: 9) Find the center, vertices and foci of each hyperbola. Plot the co-vertices and draw the asymptotes of each hyperbola. y x a) = 1 b) 5x 16y = ) Find an equation for these hyperbolas. a) 4x y + 16x + 6y + 3 = 0 b) center (3, -1); vertex (6, -1); focus (8, -1) Mixed Review: 11) What would be true about a circle, and ellipse and an hyperbola that intersect symmetrically? 1) A(n) is the collection of all points in the plane such that the distance from each point to a fixed point equals its distance to a fixed line. 13) A(n) is the collection of all points in the plane whose sum of the distances from two fixed points is a constant. 14) A(n) is the collection of all points in the plane whose difference of the distances from two fixed points is a constant. 15) For an ellipse, the foci lie on the axis; for a hyperbola, the foci lie on the axis. 16) For the ellipse x y + = 1, the major axis is along the ) Name two ways the standard form of a parabola differs from all other conic sections.

3 True or False: 18) On a parabola the distance from any point to the focus equals the distance from that point to the directrix. 19) The foci of an ellipse lie on its major axis. 0) The foci of a hyperbola lie on its transverse axis. 1) Hyperbolas always have asymptotes, and ellipses never have asymptotes. ) The equation ax y y = 0 defines an ellipse if a > 0. 3) The eccentricity of an ellipse ranges between -1 and 1. In problems #4-33 identify the equation. If it s a parabola, give its vertex, focus and directrix. If it s an ellipse, give its center, vertices and foci; and if it s a hyperbola, give its center, vertices, and foci. Graph each equation. 4) y = 16x 5) x 5 y = 1 6) y x + = ) x + 4y = 4 8) 4x y = 8 9) x 4x = y 30) y y x x = 4

4 31) 4x + 9y 16x 18y = 11 3) 4x 16x + 16y + 3 = 0 33) 9x + 4y 18x + 8y = 3 In problems #34-37, obtain an equation of the conic described. Graph the equation by hand. 34) Parabola: focus at (-,0); directrix is the line x =. 35) Hyperbola: center at (0,0); focus at (0,4); vertex at (0,-). 36) Ellipse: foci at (-3,0) and (3,0); vertex at (4,0). 37) Parabola: vertex at (,-3); focus at (, -4). Context Problems 38) A brick archway leading into your garden has a parabolic shape and is 14 feet high at the vertex. Two feet below the vertex, the archway is 6 feet wide. You have recently purchased a gigantic fountain whose circular base is 1 feet in diameter and sides are 7 feet high. Can you pass it through the archway? Why or why not? 39) You are building a wading pool that is in the shape of an ellipse. The architect has given you an equation for the pool (measure in feet) that she says would allow anyone with a precalculus background to map out the perimeter of the pool. Sketch the dimensions of the pool if the equation is 169x + 361y = Will you have to move a tree that is planted 15 feet from the center of the pool, at a 45 angle to the major axis? Why or why not? 40) Halley s comet has an elliptical orbit around the sun with an eccentricity of e = The length of the major axis is approximately 33.9 astronomical units. How close does Halley s comet come to the sun? Write an equation in standard form for the orbit of Halley s comet.

5 Other Interesting Conics Problems 41) Show that the asymptotes of the hyperbola x a y = 1 are perpendicular to each other. a 4) Find a number k such that (-, 1) is on the graph of 3x + ky = 4. Find the foci of this conic. 43) Translate the hyperbola defined by the equation 7x 5y = 48 0y 14x down 5 units and to the left 4 units and write the resulting equation in standard form. 44) Find the standard form equation of an ellipse through the point (4,) with foci at (1, -1) and (1, 5). 45) Write the equation of the hyperbola that passes through (4, ) and has asymptotes with equations y = x and y = -x ) The latus rectum of a parabola is the line segment through the focus that is perpendicular to the axis and has endpoints on the parabola. The length of the latus rectum is 4 p units long. Write the equation of a parabola with vertex at (-, 1), and the endpoints of the latus rectum are (0, 5) and (0, -3). 47) True or False. The distance from a focus of an ellipse to either endpoint of the minor axis is half the length of the major axis. Justify your answer.