Coordinate Geometry Ellipse

Please help me solve this (Doh)

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 a^{2}/c) note (e = c / a)

Thanks in advance for the reply ^^

Re: Coordinate Geometry Ellipse

i think its not the midpoint

Re: Coordinate Geometry Ellipse

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Re: Coordinate Geometry Ellipse

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**mventurina1** Please help me solve this (Doh)

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 a^{2}/c) note (e = c / a)

Thanks in advance for the reply ^^

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)

Re: Coordinate Geometry Ellipse

Re: Coordinate Geometry Ellipse

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**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 :)

Re: Coordinate Geometry Ellipse

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**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.

Re: Coordinate Geometry Ellipse

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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?

Re: Coordinate Geometry Ellipse

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**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!

Re: Coordinate Geometry Ellipse

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**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 !