Coordinate Geometry Ellipse

• Mar 23rd 2014, 06:41 AM
mventurina1
Coordinate Geometry Ellipse

Q: is the midpoint from directrix to C the focus?

determine the equation and graph of the ellipse w/ center at (4,-5), major axis vertical. eccentricity=√21/5 distance between directrices=50√21/21

(note: distance from C to a directrix = a/e or a2/c) note (e = c / a)

• Mar 23rd 2014, 06:47 AM
mventurina1
Re: Coordinate Geometry Ellipse
i think its not the midpoint
• Mar 23rd 2014, 08:27 AM
bjhopper
Re: Coordinate Geometry Ellipse
Do you mean parabola?
• Mar 24th 2014, 12:03 AM
earboth
Re: Coordinate Geometry Ellipse
Quote:

Originally Posted by mventurina1

Q: is the midpoint from directrix to C the focus?

determine the equation and graph of the ellipse w/ center at (4,-5), major axis vertical. eccentricity=√21/5 distance between directrices=50√21/21

(note: distance from C to a directrix = a/e or a2/c) note (e = c / a)

Good morning,

1. Let d denotes the distance from C to the directrix. Then you know $d = \frac{25}{21} \cdot \sqrt{21}$

With $a = e \cdot d$ follows: $a = \frac{\sqrt{21}}{5} \cdot \frac{25}{21} \cdot \sqrt{21} = 5$

2. Since $\frac{a^2-b^2}{a^2} = \frac{21}{25}~\implies~b = 2$

3. Therefore the equation of the ellipse is:

$\frac{(x-4)^2}4+\frac{(y+5)^2}{25}=1$

4. The coordinates of the foci are $F_1\left(4\ ,\ -5+\sqrt{21} \right)$ and $F_2\left(4\ ,\ -5-\sqrt{21} \right)$

5. The equation of the upper directrix is $d:y = -5+\frac{25}{21} \cdot \sqrt{21}$

6. The major axis is placed on the straight line a (see attachment). d crosses a in the point $D\left(4, -5+\frac{25}{21} \cdot \sqrt{21} \right)$

7. The midpoint between C and D is $M \left(4, -5+\frac{25}{42} \cdot \sqrt{21} \right)$ which is definitely not the focus F (compare the result at #4)
• Mar 24th 2014, 04:30 AM
mventurina1
Re: Coordinate Geometry Ellipse
ellipse :)
• Mar 24th 2014, 04:32 AM
mventurina1
Re: Coordinate Geometry Ellipse
Quote:

Originally Posted by earboth
Good morning,

1. Let d denotes the distance from C to the directrix. Then you know $d = \frac{25}{21} \cdot \sqrt{21}$

With $a = e \cdot d$ follows: $a = \frac{\sqrt{21}}{5} \cdot \frac{25}{21} \cdot \sqrt{21} = 5$

2. Since $\frac{a^2-b^2}{a^2} = \frac{21}{25}~\implies~b = 2$

3. Therefore the equation of the ellipse is:

$\frac{(x-4)^2}4+\frac{(y+5)^2}{25}=1$

4. The coordinates of the foci are $F_1\left(4\ ,\ -5+\sqrt{21} \right)$ and $F_2\left(4\ ,\ -5-\sqrt{21} \right)$

5. The equation of the upper directrix is $d:y = -5+\frac{25}{21} \cdot \sqrt{21}$

6. The major axis is placed on the straight line a (see attachment). d crosses a in the point $D\left(4, -5+\frac{25}{21} \cdot \sqrt{21} \right)$

7. The midpoint between C and D is $M \left(4, -5+\frac{25}{42} \cdot \sqrt{21} \right)$ which is definitely not the focus F (compare the result at #4)

thank you sir for the answer !!! now I understand :)
• Mar 24th 2014, 05:47 AM
earboth
Re: Coordinate Geometry Ellipse
Quote:

Originally Posted by mventurina1
thank you sir for the answer !!! now I understand :)

Hello,

I noticed to late that there is a second solution.

There exists a second directrix whose equation is
$d:y = -5-\frac{25}{21} \cdot \sqrt{21}$

This directrix crosses the straight line a in $P\left(4, -5-\frac{25}{21} \cdot \sqrt{21} \right)$

The midpoint between F and P is indeed C. So the statement in the question is true under certain conditions.
• Mar 24th 2014, 07:11 PM
mventurina1
Re: Coordinate Geometry Ellipse
Quote:

Originally Posted by earboth
Hello,

I noticed to late that there is a second solution.

There exists a second directrix whose equation is
$d:y = -5-\frac{25}{21} \cdot \sqrt{21}$

This directrix crosses the straight line a in $P\left(4, -5-\frac{25}{21} \cdot \sqrt{21} \right)$

The midpoint between F and P is indeed C. So the statement in the question is true under certain conditions.

you mean under certain conditions . the focus is the midpoint of the directrix and center?
• Mar 25th 2014, 03:31 AM
earboth
Re: Coordinate Geometry Ellipse
Quote:

Originally Posted by mventurina1
you mean under certain conditions . the focus is the midpoint of the directrix and center?

Hello,

please do me a favor and forget my last answer - it is wrong. I must have had a brain blackout. Sorry for the confusion!
• Mar 25th 2014, 10:28 PM
mventurina1
Re: Coordinate Geometry Ellipse
Quote:

Originally Posted by earboth
Hello,

please do me a favor and forget my last answer - it is wrong. I must have had a brain blackout. Sorry for the confusion!

Sure and thanks again Sir !