equation of ellipses

Question

I was given the pionts (-1,-1) (-1,5) (-3,2) (1,2) and i have to fins the equation.

Answer 1

 

1. Looking at the first two points, (-1, -1) and (-1, 5),

    the vertical line segment connecting those two is 6 units long.

 

2. Looking at the last two points, (-3, 2) and (1, 2),

    the horizontal line segment connecting those two is 4 units long.

 

Based on #1 and #2 above,

we can think of the zero-centered ellipse

of which two vertices on the major axis (y-axis in this case) are 6 units apart and

two vertices on the minor axis (x-axis in this case) are 4 units apart,

which is given by

 

     x²/2² + y²/3² = 1    (Eq. 1)

 

Two line segments from #1 and #2 above cross at the point (-1, 2),

which is the center of the ellipse we are trying to find out.

So, if we slide the zero-centered ellipse given by Eq. 1 above  

by 1 unit toward the negative direction along the x-axis and then

by 2 units toward the positive direction along the y-axis,

we get the equation of the ellipse as follows:

 

     (x+1)²/2² + (y-2)²/3² = 1

 

which is the final answer to the problem.