How would you find the foci and vertices of the following ellipse: (2x^2)/15+(8y^2)/45-(2sqrt3)/45xy=1?

Kelton Bailey 2022-10-01 Answered
Finding the foci and vertices of an ellipse.
How would you find the foci and vertices of the following ellipse: 2 x 2 15 + 8 y 2 45 2 3 45 x y = 1 ?
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Answers (2)

Matteo Estes
Answered 2022-10-02 Author has 9 answers
Step 1
Notice first of all that your ellipse is centered at O = ( 0 , 0 ) (because if (x,y) belongs to the ellipse, then also (-x, -y) belongs to it).
To find the axes of the ellipse, notice that if P = ( x , y ) is a vertex, then the tangent at P is perpendicular to PO, that is y ( y / x ) = 1. You can compute y′ by differentiating the equation of the ellipse:
4 15 x + 16 45 y y 2 3 45 y 2 3 45 x y = 0 ,
whence: y = 2 3 y 12 x 16 y 2 3 x .
Step 2
The above condition y ( y / x ) = 1 implies then that the coordinates of a vertex are related by:
y x = 1 ± 2 3 .
Plugging that into the ellipse equation you can get the coordinates of the vertices, and then of course those of the foci.
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seguitzla
Answered 2022-10-03 Author has 4 answers
Step 1
Consider a rotation: x = X cos α Y sin α , y = X sin α + Y cos α
Then your ellipse becomes 6 ( X cos α Y sin α ) 2 + 8 ( X sin α + Y cos α ) 2 2 3 ( X cos α Y sin α ) ( X sin α + Y cos α ) = 45
The term in XY has coefficient 4 cos α sin α 2 3 cos 2 α + 2 3 sin 2 α which you want to be vanishing; dividing by cos 2 α we get 2 3 tan 2 α + 4 tan α 2 3 = 0 so tan α = 2 + 4 2 3 = 1 3 or tan α = 2 4 2 3 = 3
Step 2
So we can take α = π / 6 and the equation of the ellipse becomes
X 2 ( 6 cos 2 α + 8 sin 2 α 2 3 cos α sin α ) + Y 2 ( 6 sin 2 α + 8 cos 2 α + 2 3 cos α sin α ) = 45
that is 5 X 2 + 9 Y 2 = 45 or X 2 9 + Y 2 5 = 1.
Find foci and vertices, then use the inverse rotation.
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