Conics foci (1 Viewer)

Aysce

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Q. {(x+1)^2}/8 + {(y-3)^2}/4 = 1 Find the foci and directrices. I dont understand how they got the foci.
 

Carrotsticks

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Q. {(x+1)^2}/8 + {(y-3)^2}/4 = 1 Find the foci and directrices. I dont understand how they got the foci.
This is an ellipse that has been shifted to the left by 1 unit, and up 3 units.

We notice that this has the same focal length as the exact same ellipse, but centred at the origin.

So what I'll do is shift that ellipse back to the origin, find the foci, then shift the ellipse back to where it was originally (which will 'carry' the foci along with it).
 

Carrotsticks

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Here's some working out if you're having trouble:



Oh, and the directrices are found the same way. However this time, we need not shift it 'up' by 3 units, since the directrices are vertical lines anyway, so shifting them up or down will make no difference.
 

SpiralFlex

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Drawing a pretty diagram may help.

Our new centre of the ellipse was originally now it's




For eccentricity,






The coordinates of our ORIGINAL focii were



Since there was a shift of one units to the left, and three units up.






Our original directrices were




Shifting this left, since this only affects the abssicas not the ordinates.



 

Carrotsticks

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Spiral, you could have found the coordinates of the foci using the formula:



This automatically eliminates any need to find the eccentricity.
 

SpiralFlex

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Spiral, you could have found the coordinates of the foci using the formula:



This automatically eliminates any need to find the eccentricity.
The source Ace is using requires him to use the eccentricity method for practice.
 

Aysce

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Thank you guys, so much! Do you guys mind helping me with another question?
 

Aysce

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<3

4b. Write the equation of the locus of a point P that moves such that its distance from (2,1) is half as its distance from the lie x+y-2=0.
 

Carrotsticks

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<3

4b. Write the equation of the locus of a point P that moves such that its distance from (2,1) is half as its distance from the lie x+y-2=0.
Let P be defined by the point (x,y), and define the point Q(2,1). Let the perpendicular distance between P and the line x+y-2=0 be l.



This is a bit messy, but the equation should be a rotated ellipse.

You can leave it like this, or you can expand fully and add/subtract like terms etc.
 

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