Answer to Question #221983 in Analytic Geometry for Celia

Question #221983

16x2+4y2+32x-16y-32=0

center,foci, ends of major and minor axis, ends of latus rectum


1
Expert's answer
2021-08-02T16:46:26-0400

Consider the equation 16x2+4y2+32x16y32=016x^2+4y^2+32x-16y-32=0

16(x2+2x)+4(y24y)=3216(x^2+2x)+4(y^2-4y)=32

16(x2+2x+11)+4(y24y+44)=3216(x^2+2x+1-1)+4(y^2-4y+4-4)=32

16(x2+2x+1)16+4(y24y+4)16=3216(x^2+2x+1)-16+4(y^2-4y+4)-16=32

16(x+1)2+4(y2)2=32+16+1616(x+1)^2+4(y-2)^2=32+16+16

16(x+1)2+4(y2)2=6416(x+1)^2+4(y-2)^2=64

Dividing by 64, we get

(x+1)24+(y2)216=1\frac{(x+1)^2}{4}+\frac{(y-2)^2}{16}=1

Observe that the above equation is of the form

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

So, the given equation is ellipse

Center of the ellipse =(h,k)=(1,2)=(h,k)=(-1,2)

And a2=4,b2=16a^2=4,b^2=16

Observe that a2<b2a^2<b^2

To find the foci of the ellipse, find cc using the relation c2=b2a2c^2=b^2-a^2

c2=164=12c^2=16-4=12

c=±23\Rightarrow c=\pm 2\sqrt{3}

Foci of the ellipse are: (h,kc),(h,k+c)=(1,223),(1,2+23)(h,k-\sqrt{c}),(h,k+\sqrt{c})=(-1,2-2\sqrt{3}),(-1,2+2\sqrt{3})

Ends of the major axis are: (h,kb),(h,k+b)=(1,24),(1,2+4)=(1,2),(1,6)(h,k-b),(h,k+b)=(-1,2-4),(-1,2+4)=(-1,-2),(-1,6)

Ends of the minor axis are:

(ha,k),(h+a,k)=(12,2),(1+2,2)=(3,2),(1,2)(h-a,k),(h+a,k)=(-1-2,2),(-1+2,2)=(-3,2),(1,2)

Ends of the latus rectum are:

(a2b,c),(a2b,c)=(1,23),(1,23)(\frac{-a^2}{b},c),(\frac{a^2}{b},c)=(-1,2\sqrt{3}),(1,2\sqrt{3})


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