Answer:
Step-by-step explanation:
The standard form for an ellipse is
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex] if it is horizontally stretched, or
[tex]\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1[/tex] if it is vertically stretched. This is because a is the bigger value of the 2 ALWAYS in an ellipse, so whichever term the a is under is the longer (or main) axis. If we plot the given points on a graph, all those points lie on the horizontal line y = 3. And since the focal points always lie on the main axis, this is a horizontally stretched ellipse.
The only thing we have to work with is the focus and its coordinates. The formula for a focus is:
[tex]c^2=a^2-b^2[/tex] where c is the distance from the focus to the center. Again, since this is a horizontally stretched ellipse with its main axis being horizontal, then a is defined as the distance between the center and the vertex. Counting from the center to the vertex is 8 units. So a = 8. The distance from the center to the focus is 2 times the square root of 14, so filling in our focus formula:
[tex](2\sqrt{14})^2=8^2-b^2[/tex] which simplifies to
[tex]56=64-b^2[/tex] so
[tex]b^2=8[/tex]
Now we need to talk about the h and k values. Those are the horizontal and vertical shifts as measured from the origin. Since the center is 5 units to the right of the origin, our horizontal shift is (x - 5); since the vertical shift is 3 units up from the origin, our vertical shift is (x - 3). Putting all that together now gives us:
[tex]\frac{(x-5)^2}{64}+\frac{(y-3)^2}{8}=1[/tex] which is choice C.
