Answer to Question #219520 in Analytic Geometry for Jessy

Question #219520

Write the equation of the ellipse which satisfies the given conditions.

1a. Center (2, 3), horizontal axis 8, vertical axis 4

b. Center (1, -2), horizontal major axis 8, eccentricity

2a. Foci (-2,1) and (4, 1), major axis 10

b. Foci (-3,0) and (-3, 4), minor axis 6

3a. Foci (-2,2) and (4,2), eccentricity

b. Foci (+2, 0), directrices x = +8

4a. Focus (0,0), vertex (5,0), eccentricity 0.5

b. Focus (0,0), vertex (0, 2), eccentricity 0.6

5a. Center (1,3), V(1,-1), and passing through the origin

b. Center (1, 1), V(3, 1), and passing through the origin


1
Expert's answer
2021-07-23T08:34:06-0400

1.

a)

C(2,3),2a=8,2b=4C(2, 3), 2a=8, 2b=4


(x2)2(8/2)2+(y3)2(4/2)2=1\dfrac{(x-2)^2}{(8/2)^2}+\dfrac{(y-3)^2}{(4/2)^2}=1

(x2)216+(y3)24=1\dfrac{(x-2)^2}{16}+\dfrac{(y-3)^2}{4}=1

b)

C(1,2),2a=8,ε=ca=a2b2a=32C(1, -2), 2a=8, \varepsilon=\dfrac{c}{a}=\dfrac{\sqrt{a^2-b^2}}{a}=\dfrac{\sqrt{3}}{2}


a2b2=34a2a^2-b^2=\dfrac{3}{4}a^2

b=a2=2b=\dfrac{a}{2}=2

(x1)216+(y+2)24=1\dfrac{(x-1)^2}{16}+\dfrac{(y+2)^2}{4}=1



2.

a)

F1(2,1),F2(4,1),2a=10F_1(-2,1), F_2(4, 1), 2a=10

a=5a=5

k=1,hc=2,h+c=4k=1, h-c=-2, h+c=4

C(1,1)C(1, 1)


c=a2b2=3c=\sqrt{a^2-b^2}=3

b=4b=4

(x1)225+(y1)216=1\dfrac{(x-1)^2}{25}+\dfrac{(y-1)^2}{16}=1


b)

F1(3,0),F2(3,4),2b=6F_1(-3,0), F_2(-3, 4), 2b=6

b=3b=3

h=3,kc=0,k+c=4h=-3, k-c=0, k+c=4

C(3,2)C(-3, 2)


c=a2b2=2c=\sqrt{a^2-b^2}=2

a=13a=\sqrt{13}

(x+3)29+(y2)213=1\dfrac{(x+3)^2}{9}+\dfrac{(y-2)^2}{13}=1


3.

a)

F1(2,2),F2(4,2),ca=35F_1(-2,2), F_2(4, 2), \dfrac{c}{a}=\dfrac{3}{5}

k=2,hc=2,h+c=4k=2,h-c=-2, h+c=4

C(1,2)C(1, 2)


c=a2b2=3c=\sqrt{a^2-b^2}=3

a=3(35)=5,b=5232=4a=3(\dfrac{3}{5})=5, b=\sqrt{5^2-3^2}=4


(x1)225+(y2)216=1\dfrac{(x-1)^2}{25}+\dfrac{(y-2)^2}{16}=1


b)

F(±2,0),x=±8F(\pm2,0), x=\pm8

k=0,k=0,

h+ae=2,hae=2h+ae=2, h-ae=-2

h+ae=8,hae=8h+\dfrac{a}{e}=8, h-\dfrac{a}{e}=-8

h=0h=0

a=8e,8e2=2a=8e, 8e^2=2

e=ca=12,a=4,c=a2b2=2e=\dfrac{c}{a}=\dfrac{1}{2}, a=4, c=\sqrt{a^2-b^2}=2

C(0,0)C(0, 0)


b=23b=2\sqrt{3}


x216+y212=1\dfrac{x^2}{16}+\dfrac{y^2}{12}=1


4.

a)

F1(0,0)F_1(0, 0)

V1(5,0)V_1(5,0)

e=0.5e=0.5


k=0,k=0,

h+ae=0,h+a=5h+ae=0, h+a=5


h+0.5a=0,h+a=5h+0.5a=0, h+a=5

a=10,h=5a=10, h=-5

or


hae=0,h+a=5h-ae=0, h+a=5

h0.5a=0,h+a=5h-0.5a=0, h+a=5

a=103,h=53a=\dfrac{10}{3}, h=\dfrac{5}{3}




a=10,h=5a=10, h=-5

C(5,0)C(-5,0)


c=a2b2=5c=\sqrt{a^2-b^2}=5

b=53b=5\sqrt{3}

(x+5)2100+y275=1\dfrac{(x+5)^2}{100}+\dfrac{y^2}{75}=1





a=103,h=53a=\dfrac{10}{3}, h=\dfrac{5}{3}


C(53,0)C(\dfrac{5}{3},0)

c=a2b2=53c=\sqrt{a^2-b^2}=\dfrac{5}{3}

b=533b=\dfrac{5\sqrt{3}}{3}

(x53)2100/9+y275/9=1\dfrac{(x-\dfrac{5}{3})^2}{100/9}+\dfrac{y^2}{75/9}=1




b)

F1(0,0)F_1(0, 0)

V1(0,2)V_1(0,2)

e=0.6e=0.6

h=0,h=0,k+ae=0,k+a=2k+ae=0, k+a=2




k+0.6a=0,k+a=2k+0.6a=0, k+a=2

a=5,k=3a=5, k=-3

or


kae=0,k+a=2k-ae=0, k+a=2

k0.6a=0,k+a=2k-0.6a=0, k+a=2

a=54,k=34a=\dfrac{5}{4}, k=\dfrac{3}{4}





a=5,k=3a=5, k=-3

C(0,3)C(0,-3)

c=a2b2=3c=\sqrt{a^2-b^2}=3

b=4b=4

x216+(y+3)225=1\dfrac{x^2}{16}+\dfrac{(y+3)^2}{25}=1





a=54,k=34a=\dfrac{5}{4}, k=\dfrac{3}{4}



C(0,34)C(0,\dfrac{3}{4})

c=a2b2=34c=\sqrt{a^2-b^2}=\dfrac{3}{4}

b=1b=1

x21+(y34)225/16=1\dfrac{x^2}{1}+\dfrac{(y-\dfrac{3}{4})^2}{25/16}=1






5.

a)

Center (1,3)(1, 3)

Vertices (1,1)(1, -1)


(xh)2b2+(yk)2a2=1\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1

Center (h,k)(h, k)

Vertices (h,k±a)(h, k\pm a)


h=1,k=3h=1, k=3

3a=1=>a=43-a=-1=>a=4

Passing through the origin


(01)2b2+(03)242=1\dfrac{(0-1)^2}{b^2}+\dfrac{(0-3)^2}{4^2}=1

b2=16/7b^2=16/7

(x1)216/7+(y3)216=1\dfrac{(x-1)^2}{16/7}+\dfrac{(y-3)^2}{16}=1

b)

Center (1,1)(1, 1)

Vertices (3,1)(3, 1)

(xh)2a2+(yk)2b2=1\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1

Center (h,k)(h, k)

Vertices (h±a,k)(h\pm a, k)

h=1,k=1h=1, k=1

1+a=3=>a=21+a=3=>a=2

Passing through the origin


(01)222+(01)2b2=1\dfrac{(0-1)^2}{2^2}+\dfrac{(0-1)^2}{b^2}=1

b2=4/3b^2=4/3

(x1)24+(y1)24/3=1\dfrac{(x-1)^2}{4}+\dfrac{(y-1)^2}{4/3}=1




Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment