Chapter 4 - 4.4 - Translations of Conics - 4.4 Exercises - Page 348: 46

Center: $(6,-7)$ Foci: $(6,-7+2\sqrt 3)~~and~~(6,-7-2\sqrt 3)$ Vertices: $(6,-2)~~and~~(6,-11)$

Work Step by Step

The standard form of the equation of the elipse when the major axis is: - horizontal: $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ in which $(h,k)$ is the center and $2a$ is the major axis length and $2b$ is the minor axis length. - vertical: $\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$ in which $(h,k)$ is the center and $2a$ is the major axis length and $2b$ is the minor axis length. $\frac{(x-6)^2}{4}+\frac{(y+7)^2}{16}=1$ $\frac{(x-6)^2}{2^2}+\frac{(y+7)^2}{4^2}=1$ $\frac{(x-6)^2}{2^2}+\frac{[y-(-7)]^2}{4^2}=1$ $a=4,~~b=2$ $c^2=a^2-b^2=16-4=12$ $c=2\sqrt 3$ Center: $(6,-7)$ The major axis is vertical: - the foci: $(6,-7+2\sqrt 3)~~and~~(6,-7-2\sqrt 3)$ - the vertices: $(6,-6+4)=(6,-2)~~and~~(6,-7-4)=(6,-11)$

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