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+sqrt정리), pointsize=20, rgbcolor=(0,0,1));

v2=point((-2,1-sqrt정리), 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 : .