1. 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)

2. Re: Coordinate Geometry Ellipse

i think its not the midpoint

3. Re: Coordinate Geometry Ellipse

Do you mean parabola?

4. Re: Coordinate Geometry Ellipse

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 $\displaystyle d = \frac{25}{21} \cdot \sqrt{21}$

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

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

3. Therefore the equation of the ellipse is:

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

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

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

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

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

ellipse

6. Re: Coordinate Geometry Ellipse

Originally Posted by earboth
Good morning,

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

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

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

3. Therefore the equation of the ellipse is:

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

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

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

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

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

thank you sir for the answer !!! now I understand

7. Re: Coordinate Geometry Ellipse

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
$\displaystyle d:y = -5-\frac{25}{21} \cdot \sqrt{21}$

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

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

8. Re: Coordinate Geometry Ellipse

Originally Posted by earboth
Hello,

I noticed to late that there is a second solution.

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

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

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?

9. Re: Coordinate Geometry Ellipse

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!

10. Re: Coordinate Geometry Ellipse

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 !