# [SOLVED] Ellipse - Focus and Directrix

• Feb 24th 2008, 11:00 AM
chrozer
[SOLVED] Ellipse - Focus and Directrix
An ellipse can be determined by means of a focus and a directrix. In the diagram, the directrix of an ellipse is the line $\displaystyle x = \frac {25}{3}$ and a focus is $\displaystyle (3,0)$. The relationship determining the ellipse is $\displaystyle PF = \frac {3}{5}PD$ for all point P.)

a. Use the distance formula to write and expression for $\displaystyle PF$, and set it equal to $\displaystyle \frac {3}{5}PD$. (Hint: You should not need the distance formula for $\displaystyle PD$.

b.Simply the equation you wrote in part (a), and put it in the graphing form of an equation of an ellipse. What is the length of the minor axis of this ellipse.

I posted the picture of the diagram below made in paint as best as it resembles on the paper. Ok...so my question is how do you solve this? I know the equation for and ellipse is $\displaystyle \frac {(x-h)^2 }{a^2} + \frac {(y-k)^2}{b^2} = 1$. I have know idea though what a directrix is as I haven't learned it yet.
• Feb 24th 2008, 04:57 PM
topsquark
Quote:

Originally Posted by chrozer
An ellipse can be determined by means of a focus and a directrix. In the diagram, the directrix of an ellipse is the line $\displaystyle x = \frac {25}{3}$ and a focus is $\displaystyle (3,0)$. The relationship determining the ellipse is $\displaystyle PF = \frac {3}{5}PD$ for all point P.)

a. Use the distance formula to write and expression for $\displaystyle PF$, and set it equal to $\displaystyle \frac {3}{5}PD$. (Hint: You should not need the distance formula for $\displaystyle PD$.

b.Simply the equation you wrote in part (a), and put it in the graphing form of an equation of an ellipse. What is the length of the minor axis of this ellipse.

I posted the picture of the diagram below made in paint as best as it resembles on the paper. Ok...so my question is how do you solve this? I know the equation for and ellipse is $\displaystyle \frac {(x-h)^2 }{a^2} + \frac {(y-k)^2}{b^2} = 1$. I have know idea though what a directrix is as I haven't learned it yet.

a) $\displaystyle PF = \sqrt{(x - 3)^2 + y^2} = \frac{3}{5} \left ( \frac{25}{3} - x \right )$
(The directrix is simply a line associated with the definition of the ellipse, nothing more.)

You should be able to work with that to get b) and c). Give it a try and post your attempt if you have trouble.

-Dan
• Feb 24th 2008, 06:50 PM
chrozer
Quote:

Originally Posted by topsquark
a) $\displaystyle PF = \sqrt{(x - 3)^2 + y^2} = \frac{3}{5} \left ( \frac{25}{3} - x \right )$
(The directrix is simply a line associated with the definition of the ellipse, nothing more.)

You should be able to work with that to get b) and c). Give it a try and post your attempt if you have trouble.

-Dan

Ok I simplified it and got $\displaystyle \frac {x^2}{25} + \frac {y^2}{16} = 1$, with the minor axis being 4. Is this right? Seems like it's right.
• Feb 24th 2008, 07:49 PM
topsquark
Quote:

Originally Posted by chrozer
Ok I simplified it and got $\displaystyle \frac {x^2}{25} + \frac {y^2}{16} = 1$, with the minor axis being 4. Is this right? Seems like it's right.

That is correct. Good job!

-Dan
• Feb 25th 2008, 05:09 AM
chrozer
Quote:

Originally Posted by topsquark
That is correct. Good job!

-Dan

Thnx so much. :D