SOLUTION: Write the equation for the ellipse in standard form and general form. center (3,-2), passing through (-4,-2), (10,-2), (3,1), and (3,-5)

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Write the equation for the ellipse in standard form and general form. center (3,-2), passing through (-4,-2), (10,-2), (3,1), and (3,-5)      Log On


   

Question 1204632: Write the equation for the ellipse in standard form and general form.
center (3,-2), passing through (-4,-2), (10,-2), (3,1), and (3,-5)
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Write the equation for the ellipse in standard form and general form.
%28x-h%29%5E2%2Fa%5E2+%2B%28y-k%29%5E2%2Fb%5E2=1
given:
center (3,-2)=(h,k)

so far, equation is:
%28x-3%29%5E2%2Fa%5E2+%2B%28y%2B2%29%5E2%2Fb%5E2=1

passing through (-4,-2), (10,-2), (3,1), and (3,-5)
use two points to calculate a and b
(3,1)
%283-3%29%5E2%2Fa%5E2+%2B%281%2B2%29%5E2%2Fb%5E2=1
%280%29%5E2%2Fa%5E2+%2B3%5E2%2Fb%5E2=1
9%2Fb%5E2=1
9=b%5E2
b=3

and (10,-2)

%2810-3%29%5E2%2Fa%5E2+%2B%28-2%2B2%29%5E2%2F9=1
7%5E2%2Fa%5E2+%2B%280%29%5E2%2F9=1
49%2Fa%5E2+=1
a%5E2+=49
a=7

equation is:
%28x-3%29%5E2%2F49+%2B%28y%2B2%29%5E2%2F9=1





Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The center is given as (3,-2).

Two points on the ellipse are (-4,-2) and (10,-2); each of those is 7 units horizontally from the center.

Two other points are (3,1) and (3,-5); each of those is 3 units vertically from the center.

So the ellipse has a horizontal semi-major axis of length 7 and a vertical semi-minor axis of length 3.

Given that information, the equation in standard form is

%28x-3%29%5E2%2F7%5E2%2B%28y%2B2%29%5E2%2F3%5E2=1

%28x-3%29%5E2%2F49%2B%28y%2B2%29%5E2%2F9=1

Convert to general form by multiplying by (9*49=441), expanding the expressions in parentheses, and simplifying.

9%28x-3%29%5E2%2B49%28y%2B2%29%5E2=441
9%28x%5E2-6x%2B9%29%2B49%28y%5E2%2B4y%2B4%29=441
9x%5E2-54x%2B81%2B49y%5E2%2B196y%2B196=441
9x%5E2%2B49y%5E2-54x%2B196y=164 or 9x%5E2%2B49y%5E2-54x%2B196y-164=0