Question

Graph the ellipse, Plot the foci of the ellipse 100pts

Answer 1

Explanation:

The general equation for an ellipse with center (h, k) is:

`\boxed{((x-h)^2)/(a^2)+((y-k)^2)/(b^2)=1}`

If a > b, the ellipse is horizontal.

If b > a, the ellipse is vertical.

Given equation:

`((x-5)^2)/(4)+((y+5)^2)/(9)=1`

As b > a, the ellipse is vertical. Therefore:

• b is the major radius and 2b is the major axis.
• a is the minor radius and 2a is the minor axis.
• Vertices = (h, k±b)
• Co-vertices = (h±a, k)
• Foci = (h, k±c) where c² = b² - a²

Comparing the given equation with the standard form, we get:

• 
  `h = 5`
• 
  `k = -5`
• 
  `a^2=4 \implies a=2`
• 
  `b^2=9 \implies b=3`

Therefore:

• 
  `\textsf{Center}= (5, -5)`
• 
  `\textsf{Major axis}=2 \cdot 3 = 6`
• 
  `\textsf{Minor axis}=2 \cdot 2 = 4`
• 
  `\textsf{Vertices:} \;\;(h, k \pm b)=(5,-5 \pm 3)=(5,-8)\;\;\textsf{and}\;\;(5,-2)`
• 
  `\textsf{Co-vertices:}\;\;(h \pm a, k)=(5 \pm 2, -5)=(3, -5)\;\; \textsf{and}\;\;(7, -5)`

To graph the ellipse:

• Plot the center at (5, -5).
• Plot the vertices at (5, -8) and (5, -2). The distance between them is the major axis.
• Plot the co-vertices at (3, -5) and (7, -5). The distance between them is the minor axis.