Ellipese and Parabolas

[geminibaby101]

1. How do you find the length of the major and minor axes of an ellipse with the equation 16x^2+25y^2+32x-150y=159 ?

Heres my work :
16x^2+25y^2+32x-150y=159
16x^2+32x+25y-150y=159
16(x^2+2x+ ?)+ 25(y^2-6y+ ?)=159 +16(?)+25(?)
16(x^2+2x+1)+25(y^2-6y+9)=159+16+225
**Everytime i try to solve the each of these problems i got stuck. The examples in my book don't help me either.


2. How do i find the foci of an ellipse with the equation 9x^2+16y^2-18x+64y=71 ?


3.How do i find the foci of a hyperbola with the eqaution 9y^2-72y-16x^2-64x-64=0 ?

4. How do i find the asymptotes of a hyberbola with theeqaution y^2=36+4x^2 ?

5.How do i find the vertices of a hyperbola with the equation (x+3)^2-4(y-2)^2=4 ?

 

[soroban]

Hello, geminibaby101!

I'll finish #1 for you.
You've done the hard part . . . you're almost there.

1. Find the length of the major and minor axes of the ellipse: \(\displaystyle \,6x^2\,+\,25y^2\,+\,32x\,-\,150y\:=\:159\; ?\)

Here's my work:
\(\displaystyle \;\;16x^2\,+\,25y^2\,+\,32x\,-\,150y\;=\;159\)
\(\displaystyle \;\;16x^2\,+\,32x\,+\,25y\,-\,150y\;=\;159\)
\(\displaystyle \;\;16(x^2\,+2x\,+\,?)\,+\,25(y^2\,-\,6y\,+\,?)\;=\;159\,+\,16(?)\,+\,25(?)\;\) . . . I like this step!
\(\displaystyle \;\;16(x^2\,+\,2x\,+\,1)\,+\,25(y^2\,-\,6y\,+\,9)\;=\;159\,+\,16\,+\,225\)

You have: \(\displaystyle \,16(x\,+\,1)^2\,+\,25(y\,-\,3)^2\;=\;400\)

Divide by 400: \(\displaystyle \L\;\frac{(x\,+\,1)^2}{25}\,+\,\frac{(y\,-\,3)^2}{16}\;=\;1\)

Now we know all about this ellipse . . .
\(\displaystyle \;\;\)The center is: \(\displaystyle (-1,\,3)\,\) and \(\displaystyle \,a\,=\,5,\;b\,=\,4\)

The length of the major axis is \(\displaystyle 10.\)
The length of the minor axis is \(\displaystyle 8.\)