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((x+3)^(2))/(25)+((y-8)^(2))/(100)=1 W...

`((x+3)^(2))/(25)+((y-8)^(2))/(100)=1`
Which conic section does this equation define? Also find, if they exist,
(i) the center
(ii) the vertex/vertices
(iii) the focus/foci
(iv) the directrix
(v) the asymtotes
(vi) the eccentricity

Text Solution

Verified by Experts

This is an ellipes with a y-orentation since the larger denominator has y in the numberator.
In this problem `h=-3,k=8,a=10,b=5,andc=sqrt(10^(2)-5^(2))=sqrt(75)=5sqrt(3)`.
(i) The center is (-3,8).
(ii) The vertices are (-3,-2) and (-3,18)
(iii) The foci are `(-3,8+5sqrt3)and(-3,8-5sqrt3)`.
(iv) An ellipes does not have a directrix.
(v) An ellipse does not have asymptotes.
(vi) The eccentricity is `(sqrt3)/(2)`.
If an equation of a conic is not in standard form, completing the square will yield a standard- form equation. If the equation has squared terms in both x and y, completing the square in both variables will result in the standard equation of an ellipes or hyperbola. If only one of the variables is squared in the original equation, completing the square in that variable will result in the standard equation of a parabola.
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