Math Analysis Problem Set 14-01 1. Find the equation of the circle for which the segment from ( 6,−3) to ( −8,12 ) is a diameter. 2. Find the equation of any circle having the segment from ( 8,−2 ) to ( 4,−8 ) as a radius. 3. Find the equation of the circle x 2 + y 2 − 12x − 6y + 33 = 0 in standard form and identify its center and radius. 4. Find the equation of the circle 8x 2 + 8y 2 − 12x + 20y + 5 = 0 in standard form and identify its center and radius. 5. Find the equation of the circle passing through the points ( −4,5 ) , ( 8, 3) , and ( 4,8 ) . [There’s a video explaining how to go about this if you’re still lost after sitting down and thinking it through…sit down and think it through before watching the video!] 6. Write the equation of the ellipse with foci ( 3,−1) and ( 9,−1) and covertex ( 6, 3) . Identify all information you know about the ellipse. 7. Write the equation of the ellipse with vertex ( −1,5 ) and focus ( −1, 3) if the major axis is 11 units long. Identify all information you know about the ellipse. 8. The points ( 6,8 ) and ( 4, 7 ) are the vertex and covertex (though not necessarily in that order) of an ellipse. Write the equation of all possible ellipses satisfying this information. Find the foci and eccentricity as well. 9. Find everything about each ellipse below. a. 49x 2 + 121y 2 − 392x + 1694y + 784 = 0 b. 16x 2 + 625y 2 + 160x − 3750y − 3975 = 0 10. Find the equation of the ellipse with vertex ( 2,10 ) , focus ( 2, 7 ) , and eccentricity 4 7.
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Math Analysis Problem Set 14-02 1. Find the equation of all parabolas with focus ( −2,1) and passing through the point ( 8,−23) . [If you can’t do this quickly you’re in trouble as far as pacing yourself goes…] 2. Find the equation of the circle passing through the points ( −2,1) , ( −1, 3) , and ( 6,2 ) . 3. Find the equation of the hyperbola with asymptotes y + 5 = ± a. a vertex at (12,−5 ) . b. a vertex at ( 3,−8 ) . c. a focus at (12,−5 ) .
5 ( x − 3) and 7
4. Write the equation of the parabola with directrix x = 5 and a. vertex (12,6 ) . b. vertex ( −9,−3) . c. focus ( 7, 4 ) . 5. Write the equation of any possible ellipse having eccentricity e = 5 9 , a focus at ( 8,1) , and a vertex at (12,1) . 6. Write the equation of the ellipse whose vertices and covertices, respectively, are the foci and covertices, respectively, of the hyperbola: 49x 2 − 576y 2 + 392x + 3456y + 23824 = 0 . 7. Given the parabola y 2 − 36x + 10y + 133 = 0 , write the equation of the: a. hyperbola with covertices at the end points of the latus rectum of the parabola and vertex at the vertex of the parabola. b. ellipse with a focus at the vertex of the parabola and covertices at the end points of the latus rectum of the parabola.
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Math Analysis Problem Set 14-03 1. Write the equation of the hyperbola satisfying: 3 a. Asymptotes y − 4 = ± ( x + 3) with a vertex at ( −7, 4 ) . 5 3 b. Asymptotes y − 4 = ± ( x + 3) with a vertex at ( −3,8 ) . 5 3 c. Asymptotes y − 4 = ± ( x + 3) with a focus at ( 5, 4 ) . 5 3 d. Asymptotes y − 4 = ± ( x + 3) with a focus at ( −3,0 ) . 5 5 e. Eccentricity e = 3 with asymptotes y + 6 = ± ( x − 4 ) . 12 5 f. Eccentricity e = , with center ( −5,−4 ) and vertex ( −5, 3) . 3 2. Find all possible hyperbola with a focus at ( −9, 4 ) and a vertex at ( −10, 4 ) . 5 3. Find all possible ellipses with center ( 8,−2 ) , eccentricity e = , and tangent to the 14 y-axis. 4. Find the equation of the ellipse whose vertices and co-vertices, respectively, are the 2 2 y + 2 ) ( x − 3) ( − = 1. foci and co-vertices, respectively, of the hyperbola 16 25
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Math Analysis Problem Set 14-04 1. Given the figure at the right. a. Write the equation of the hyperbola with asymptotes that are shown and a vertex at point A. b. Write the equation of the hyperbola with asymptotes that are shown and a co-vertex at point A. c. Write the equation of the hyperbola with asymptotes that are shown and a focus at point A.
2. Given the hyperbola 25x 2 − 36y 2 − 150x − 72y − 711 = 0 , find… a. The equation of the the ellipse passing through the vertices and covertices of the given hyperbola. b. The equation of the ellipse passing though the foci and covertices of the given hyperbola. c. The equation of the ellipse circumscribed about the “asymptote box” of the given hyperbola.
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Math Analysis Problem Set 14-05 1. In each of the figures below, one or more vertex, focus, or covertex is shown for ellipse. Find the standard form of the equation and all information for each ellipse. a. b.
c.
d.
e.
f.
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2. Sketch a generic looking hyperbola. Using one vertex, which is a point on the hyperbola, and the foci, determine what the constant difference between the distances from any point on the hyperbola to the foci is in terms of a, b, and or c. 3. In each of the figures below, the foci and a point on the hyperbola are given. Find standard equation and all additional information about each. a. b.
4. Find the standard form of both possible hyperbolas who have the corners of their asymptote boxes at the given coordinates. a. ( −6, 3) , ( −6,9 ) , ( 4,9 ) , and ( 4, 3) b. (12,−4 ) , (12,20 ) , ( −8,20 ) , and ( −8,−4 )
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Math Analysis Problem Set 14-06 1. Find everything for each of the following. Show your work. Check on your calculator! a. 25x 2 + 36y 2 + 150x − 288y − 99 = 0 b. 144x 2 − 49y 2 − 288x − 294y + 6759 = 0 c. 2025x 2 − 484y 2 + 4050x − 2904y − 29556 = 0 2. Find the eccentricity of an ellipse having foci ( 5,2 ) and (11,2 ) and passing through the point ( 9,5 ) . 3. Find two different ellipses with horizontal major axis having their centers at ( 5, 3) and eccentricity e = 4 5 . 4. True or false and why? All hyperbolas, with horizontal transverse axes, having the same asymptotes also have the same eccentricity. 5. Sketch figures showing how many intersection points an ellipse and a hyperbola can have: 0, 1, 2, 3, and 4. 6. Use your calculator to find the intersect points of
( x − 2 )2 − ( y − 1)2
( x − 3)2 + ( y − 2 )2 36
16
= 1 and
= 1 . The intersection command from the graphing screen works on 4 16 conic sections but you have to use it in pretty much the same way you use Zoom Box. (Pick the top left corner, drag to the bottom right corner, and it finds the intersection that’s in the box you drew.)
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Math Analysis Problem Set 14-07 1. Write the equation of the circle passing through the points ( −3,2 ) , ( 5, 4 ) , and ( 9,1) . Show your steps. 2. Suppose that an ellipse and a hyperbola share the foci ( 5,−2 ) and ( 5,10 ) along with a covertex at ( 7, 4 ) . a. Find the equation of both conics. b. Find everything about each of the conics. 3. A parabola passes through the point ( −4,−8 ) and a focus at ( 20,−1) . Find all possible parabolas satisfying this information. 4. Find the equation of a hyperbola having asymptotes 5x − 3y = 20 and 5x + 3y = 29 with a focus at ( x,9 ) . 5. Find the area of the quadrilateral determined by the foci and vertices of .the conic section 121x 2 + 36y 2 − 726x + 144y − 3123 = 0 . 6. Find the intersections of the conics
( x − 1)2 − 16
( y + 3)
2
= 1 and
( x + 1)2 + ( y − 2 )2 49
121
=1.
This is a calculator question!
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