|
Question 1144745: A semielliptical archway is to be formed over the entrance to an estate.bthe arch is to be set on pillars that are 10 feet apart and is to have a height (atop the pillars) of 4 feet. Where should the foci be placed in order to sketch the arch?
Answer by KMST(5410)   (Show Source):
You can put this solution on YOUR website! The semi-ellipse center, and the leftmost, rightmost and top points of the arch would be represented in the sketch like this:
I placed the center of the ellipse at the origin of my system of coordinates, the point  , with the x-axis representing the line connecting the top of the pillars.
The leftmost, rightmost and top points of the arch represented by the points(-5,0) , (5,0) and (0,4) respectively.
The distances between the center of the semi-ellipse and those points are 5,5, and 4 respectively.
The larger distance is  the semimajor axis of the ellipse.
The distance from the center to the top point is  , the semi-minor axis of the ellipse.
The distance from the center to either focus,  is related to  } and  by
 <-->   .
The foci should be placed at (-3,0) and (3,0).
ABOUT ELLIPSES
Ellipses are closed curves that looks like circles stretched in one direction, like this:
Like a circle, an ellipse has a center, but the points on the ellipse are not all at the same distance from that center.
An ellipse has two axes of symmetry, perpendicular to each other, passing through the center.
Being "stretched" along the direction of one of those axes of symmetry,
the ellipse has two extreme points along that axis that are the farthest from the center.
Those points are called the vertices of the ellipse.
The points where the other axis of symmetry crosses the ellipse are called the co-vertices.
The line segment and the distance between the two vertices is called the major axis.
The distance between the center of the ellipse and a vertex is called the semi-major axis, and is usually represented by .
The line segment and the distance between the two co-vertices is called the minor axis.
The distance between the center of the ellipse and a co-vertex is called the semi-minor axis, and is usually represented by .
The foci should be to the left and right of the center, at (-c,0) and (c,0) where  .
The distance from the center of an ellipse to a co-vertex is called the semi-minor axis, and labeled as .
An ellipse is defined as the locus of the points such that the sum of their distances to the foci is a constant.
Each focus of an ellipse is between the center of the ellipse and a vertex at a distance from the center.
The distance from a vertex to the nearest focus is  , and the distance to the other focus is  ,
so the sum of distances from a vertex to the foci is  .
The distance from a co-vertex to each focus is the same and must be for the sum to be  .
The distances to the center to a co-vertex (b), from the center to a focus (c) are the legs of a right triangle whose hypotenuse is the distance a from co-vertex to focus.
| |
|
| |
