Quandaries
and Queries
Hello,
I working on a problem that asks me to give the equation of an ellipse when only the location of the directrix and the length of the latus rectum are given. No other points on the ellipse are given. Again, the only "givens" are:
Length of latus rectum = 12
Location of directrix is x = 16
If I could determine the eccentricity, I could proceed from there by taking the ratio of the distance from a focus to the latus rectum point to the distance of the point from the directrix, but I lack the x coordinate of c. I've searched the text, and feel I've "missed something" somewhere! I note that the latus rectum segment is unique in one respect in that it is parallel to the directrix, where any other line segment on the ellipse to the focus would not be. Please indicate where I'm going wrong.
Thanks.
Allan
Hi Allen,
You are probably supposed to assume that the ellipse is in standard position. That means that its equation is
x2/a2 + y2/b2 = 1.
The latus rectum is the length of the chord through the focus that is perpendicular to the major axis. The directrix has the equation
x = a2/c.
You also know that
b2 + c2 = a2.
That's 3 equations in 3 unknowns so, in theory, you can find the solution. Unless I made a mistake, the resulting equation one must solve isn't very pretty!
Chris