Question 312310: I can't seem to get these right, so thanks for helping if you can.
1.) Change the following equation into vertex form:
y = 12x^2 - 96x + 200
2.) Write an equation for the ellipse that has foci (0,-15) and (0,15) and focal constant 34. (And what exactly is a focal constant?)
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Factor out the lead coefficient on  , divide the resulting coefficient on  by 2, square the result, add that result inside of the parentheses, and compensate by subtracting the added term times the lead coefficient.
\ +\ 200\ -\ 192)
Factor the trinomial
^2\ +\ 8)
So, parabola, vertical symmetry, vertex at (4,8), focus at (4,11), directrix  .
=========================
First things first. The focal constant is the sum of the distances of the foci to the ellipse at any point -- such being the definition of an ellipse.
Your foci are at (0,-15) and (0,15). This tells us a couple of things. First, the major axis is on the x-axis and the center is at the origin.
The distance from one of the focal points to a vertex plus the distance of the other focal point to the same vertex is 34...just like the sum of the distances from the foci to any other point on the ellipse. But considering the distances along the x-axis, you can use the fact that the foci are 30 units apart. So, the distance from the near focus to a vertex is 34 - 30 divided by 2, or 2.
Now we know that the two vertices are at (0,-17) and (0,17) and we have a semi-minor axis that measures 17. But we knew that already because the focal constant is always equal to the measure of the major axis.
Next there is another relationship that also holds for every ellipse. If  is the distance from the center of the ellipse to either focus, and  is the measure of the semi-major axis (one-half of the major axis or the distance from the center of the ellipse to either vertex.) and  is the measure of the semi-minor axis (one half of the minor axis or the measure of the segment from the center of the ellipse to either endpoint of the minor axis), then the following relationship holds:
In your case, you have found  and  , so a little arithmetic tells us 
Now we know enough to create the desired equation.
An equation of an ellipse with center at ) , horizontal semi-major axis and vertical semi-minor axis is:
^2}{a^2}\ +\ \frac{(y\ -\ k)^2}{b^2}\ =\ 1)
Tossing in the numbers we know:
Center: )
Semi-major axis:
Semi-minor axis:
So:
Which can also be expressed as:
or
John
|
|
|
