9.5 Conic Section
1-4. Sketch the parabola with the given equation. Show and label its vertex, focus, axis, and directrix.
1. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-1.html
var('x,y') implicit_plot((y-2)^2==3*(x-3),(x,-10,10),(y,-10,10)).show(aspect_ratio=1, xmin=0, xmax=12, ymin=-4, ymax=8) |
¢¡ vertex : / focus : / axis : / directrix : .
2. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-2.html
x,y,a,b,c=var('x,y,a,b,c') parabola=implicit_plot(6*y+x^2==0,(x,-3,3),(y,-3,3)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3) |
¢¡ vertex : / focus : / axis : / directrix : .
3. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-3.html
x,y,a,b,c=var('x,y,a,b,c') parabola=implicit_plot((y-3)^2+2*(x+1)==0,(x,-5,1),(y,0,7)).show(aspect_ratio=1, xmin=-5, xmax=1, ymin=0, ymax=7) |
¢¡ vertex : / focus : / axis : / directrix : .
4. .
5-8. Find the vertices and foci of the ellipse and sketch its graph.
5. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-5.html
x,y,a,b,c=var('x,y,a,b,c') ellipse=implicit_plot(x^2/6+y^2/4==1,(x,-3,3),(y,-3,3)).show(aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3) |
solve(a^2==6,a) |
[a == -sqrt(6), a == sqrt(6)]
solve(b^2==4,b) |
[b == -2, b == 2]
solve(c^2==6-4,c) |
[c == -sqrt(2), c == sqrt(2)]
x,y,a,b,c=var('x,y,a,b,c') ellipse=implicit_plot(x^2/6+y^2/4==1,(x,-3,3),(y,-3,3)) f1=point((-sqrt(2),0), pointsize=20, rgbcolor=(1,0,0)); f2=point((sqrt(2),0), pointsize=20, rgbcolor=(1,0,0)); v1=point((-sqrt(6),0), pointsize=20, rgbcolor=(0,0,1)); v2=point((sqrt(6),0), pointsize=20, rgbcolor=(0,0,1)); v3=point((0,-2), pointsize=20, rgbcolor=(0,0,1)); v4=point((0,2), pointsize=20, rgbcolor=(0,0,1)); show(ellipse+f1+f2+v1+v2+v3+v4,aspect_ratio=1, xmin=-3, xmax=3, ymin=-3, ymax=3) |
6. .
7. .
8. .
9-11. Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.
9. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-9.html
x,y,a,b,c=var('x,y,a,b,c') hyperbola=implicit_plot(x^2/36-y^2/16==1,(x,-15,15),(y,-15,15)).show(aspect_ratio=1, xmin=-15, xmax=15, ymin=-15, ymax=15) |
solve(a^2==36,a) |
[a == -6, a == 6]
solve(b^2==16,b) |
[b == -4, b == 4]
solve(c^2==36+16,c) |
[c == -2*sqrt(13), c == 2*sqrt(13)]
x,y,a,b,c=var('x,y,a,b,c') hyperbola=implicit_plot(x^2/36-y^2/16==1,(x,-15,15),(y,-15,15)) f1=point((-2*sqrt(13),0), pointsize=20, rgbcolor=(1,0,0)); f2=point((2*sqrt(13),0), pointsize=20, rgbcolor=(1,0,0)); v1=point((-6,0), pointsize=20, rgbcolor=(0,0,1)); v2=point((6,0), pointsize=20, rgbcolor=(0,0,1)); show(hyperbola+f1+f2+v1+v2,aspect_ratio=1, xmin=-15, xmax=15, ymin=-15, ymax=15) |
10. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-10.html
x,y,a,b,c=var('x,y,a,b,c') hyperbola=implicit_plot(y^2/81-x^2/25==1,(x,-15,15),(y,-20,20)).show(aspect_ratio=1, xmin=-15, xmax=15, ymin=-20, ymax=20) |
solve(a^2==81,a) |
[a == -9, a == 9]
solve(b^2==25,b) |
[b == -5, b == 5]
solve(c^2==81+25,c) |
[c == -sqrt(106), c == sqrt(106)]
x,y,a,b,c=var('x,y,a,b,c') hyperbola=implicit_plot(y^2/81-x^2/25==1,(x,-15,15),(y,-20,20)) f1=point((0,-sqrt(106)), pointsize=20, rgbcolor=(1,0,0)); f2=point((0,sqrt(106)), pointsize=20, rgbcolor=(1,0,0)); v1=point((0,-9), pointsize=20, rgbcolor=(0,0,1)); v2=point((0,9), pointsize=20, rgbcolor=(0,0,1)); show(hyperbola+f1+f2+v1+v2,aspect_ratio=1, xmin=-15, xmax=15, ymin=-20, ymax=20) |
11. .
12-14. Identify the type of conic section whose equation is given and find the vertices and foci.
12. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-12.html
¢¡ ¢¡ ¢¡ : Ellipse
x,y,a,b,c=var('x,y,a,b,c') ellipse=implicit_plot(x^2/3+(y-1)^2/1==1,(x,-2,2),(y,-2,4)).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=4) |
solve(a^2==3,a) |
[a == -sqrt(3), a == sqrt(3)]
solve(b^2==1,b) |
[b == -1, b == 1]
solve(c^2==3-1,c) |
[c == -sqrt(2), c == sqrt(2)] : vertices (,1) / focus ( ,1)
var('x,y') ellipse=implicit_plot(x^2/3+(y-1)^2/1==1,(x,-2,2),(y,-2,4)) f1=point((-sqrt(2),1), pointsize=20, rgbcolor=(1,0,0)); f2=point((sqrt(2),1), pointsize=20, rgbcolor=(1,0,0)); v1=point((-sqrt(3),1), pointsize=20, rgbcolor=(0,0,1)); v2=point((sqrt(3),1), pointsize=20, rgbcolor=(0,0,1)); v3=point((0,0), pointsize=20, rgbcolor=(0,0,1)); v4=point((0,2), pointsize=20, rgbcolor=(0,0,1)); show(ellipse+f1+f2+v1+v2+v3+v4,aspect_ratio=1,xmin=-2, xmax=2, ymin=-1, ymax=3) |
13. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-13.html
¢¡ : Parabola with vertices (1,2) and focus (3,2)
var('x,y') parabola=implicit_plot((y-2)^2==8*(x-1),(x,-1,8),(y,-4,8)) f1=point((1,2), pointsize=20, rgbcolor=(1,0,0)); v1=point((3,2), pointsize=20, rgbcolor=(0,0,1)); show(parabola+f1+v1,aspect_ratio=1, xmin=-1, xmax=8, ymin=-4, ymax=8) |
14. .
http://matrix.skku.ac.kr/cal-lab/cal-9-5-14.html
¢¡ : Hyperbola with vertices and focus (-2,4), (-2,-2)
var('x,y') hyperbola=implicit_plot((y-1)^2/5-(x+2)^2/4==1,(x,-10,87),(y,-8,10)) f1=point((-2,4), pointsize=20, rgbcolor=(1,0,0)); f2=point((-2,-2), pointsize=20, rgbcolor=(1,0,0)); v1=point((-2,1+sqrtÁ¤¸®), pointsize=20, rgbcolor=(0,0,1)); v2=point((-2,1-sqrtÁ¤¸®), pointsize=20, rgbcolor=(0,0,1)); show(hyperbola+f1+f2+v1+v2,aspect_ratio=1, xmin=-10, xmax=7, ymin=-8,ymax=10) |
15-22. Find an equation for the conic that satisfies the given conditions.
15. Parabola, vertex , focus .
.
16. Parabola, vertex , focus .
.
17. Ellipse, foci , vertices .
.
18. Ellipse, foci , vertices .
.
19. Ellipse, center , focus vertex .
.
20. Hyperbola, foci , vertices .
.
21. Hyperbola, foci , asymptotes .
.
22. Hyperbola, focus , asymptotes and .
.
23-28. Write a polar equation of a conic with the focus at the origin and the given data.
23. Hyperbola, eccentricity , directrix .
.
24. Hyperbola, eccentricity , directrix .
.
25. Ellipse, eccentricity , directrix .
.
26. Ellipse, eccentricity , directrix .
.
27. Parabola, eccentricity , directrix .
.
28. Parabola, eccentricity , directrix .
.
29-32. Find the eccentricity, identify the conic, give an equation of the directrix, and sketch the conic.
29. .
, , Parabola , directrix : .
30. .
, , Parabola , directrix : .
31. .
, , Ellipse, directrix : .
32. .
, , Ellipse, directrix : .