Question 437631
4x^2+3y^2=48
<pre><font face = "consolas" color = "indigo"><b>

Get it in the form

{{{(x-h)^2/a^2 + (y-k)^2/b^2 = 1}}} if it looks like this &#4540;:

or in the form 

{{{(x-h)^2/b^2 + (y-k)^2/a^2 = 1}}} if it looks like this &#8009;
:

And the way you tell is by the fact that aČ is larger than bČ

{{{4x^2+3y^2=48}}}

Divide through by 48 to get 1 on the right:

{{{4x^2/48+3y^2/48=48/48}}}

Simplify

{{{x^2/12+y^2/16 = 1}}}

Now write x as (x-0) and y as (y-0) 

{{{(x-0)^2/12+(y-0)^2/16 = 1}}}

16 is larger than 12 so the ellipse looks like this: 0
and it is in this form:

{{{(x-h)^2/b^2+(y-k)^2/a^2 = 1}}}

Comparing directly gives:

center = (h,k) = (0,0), aČ = 16, so a = 4, bČ = 12, so b = {{{sqrt(12)=2sqrt(3)}}}

Major axis has length 2a = 2(4) = 8

So we plot the center and draw the major axis 8 units long with
the center (0,0)at its midpoint, which is the green line below 
extending from (0,-4) to (0,4):

{{{drawing(400,400,-5,5,-5,5,

graph(400,400,-5,5,-5,5), green(line(0,-4,0,4))  )}}}

Next we'll draw the minor axis 2b = {{{4sqrt(3)}}} long or about 6.93 
units long with the center (0,0) at its midpoint, which is the blue 
line below extending from ({{{-2sqrt(3)}}},0) to ({{{2sqrt(3)}}},0), 
which is about the points (-1.73,0) to (1.73)

{{{drawing(400,400,-5,5,-5,5, blue(line(-2sqrt(12),.05,2sqrt(12),.05),line(-1.7320509,-.05,1.7320509,-.05)),
graph(400,400,-5,5,-5,5), green(line(0,-4,0,4))  )}}}

and sketch in the ellipse:

{{{drawing(400,400,-5,5,-5,5, blue(line(-2sqrt(3),.05,2sqrt(3),.05),line(-2sqrt(3),-.05,2sqrt(3),-.05)), arc(0,0,4sqrt(3),-8),
graph(400,400,-5,5,-5,5), green(line(0,-4,0,4))  )}}}

To find the focal points, we calculate c from

{{{c = sqrt(a^2-b^2) = sqrt(16-12) = sqrt(4) = 2}}}

So the focal points are on the major axis c = w units from the
center:

So they are (0,2) and (0,-2)

{{{drawing(400,400,-5,5,-5,5, blue(line(-2sqrt(3),.05,2sqrt(3),.05),line(-2sqrt(3),-.05,2sqrt(3),-.05)), arc(0,0,4sqrt(3),-8),
circle(0,2,.1), circle(0,-2,.1),
locate(0,2,"F(0,2,)"), locate(0,-2,"F(0,-2)"),
graph(400,400,-5,5,-5,5), green(line(0,-4,0,4))  )}}}


Edwin</pre>