For each:
a) Identify the ellipse as horizontal or vertical
b) Give the coordinates of
1) The center
2) The major vertices
3) The minor vertices
4) The foci
1) [(x+4)2/64] + (y2/100) = 1
2) [(x-1)2 / 36] + [(y+3)2 / 25] = 1
#1 vertical because y's denominator is greater.
center: (-4,0)
a = 10
b = 8
c = 6
I give the semi-major axes and focal distance. You can easily convert yo coordinates.
Do #2 similarly
To solve these questions and find the information requested, we'll need to understand the general equation of an ellipse and apply it to each given equation.
The standard form of an ellipse equation is
(x - h)² / a² + (y - k)² / b² = 1,
where (h, k) represents the coordinates of the ellipse center, "a" represents the distance from the center to the major axis vertices, and "b" represents the distance from the center to the minor axis vertices.
Using this equation, we can determine the orientation of the ellipse (horizontal or vertical) as well as the coordinates of the center, major vertices, minor vertices, and foci.
Let's solve for each given equation:
1) [(x+4)²/64] + (y²/100) = 1
a) Based on the equation, we can see that the denominator values 64 (under x) and 100 (under y) are the squares of the semi-major axis and semi-minor axis, respectively. Since 100 > 64, this means the ellipse is vertically oriented.
b) Now, let's determine the coordinates:
The center = (-4, 0) since the form of the equation is [(x - h)²/a² + (y - k)²/b² = 1, and (h, k) represents the center point.
Major vertices: To find the major vertices, we move a distance of ±a units from the center along the major axis. In this case, a = √64 = 8. So, the major vertices are (-4, 8) and (-4, -8).
Minor vertices: To find the minor vertices, we move a distance of ±b units from the center along the minor axis. In this case, b = √100 = 10. So, the minor vertices are (-4 - 10, 0) and (-4 + 10, 0), which simplifies to (-14, 0) and (6, 0).
To find the foci, we can use the equation c² = a² - b². In this case, c² = 100 - 64 = 36. Taking the square root, c = √36 = 6. Since the ellipse is vertically oriented, the foci will be at (-4, ±6).
2) [(x-1)²/36] + [(y+3)²/25] = 1
a) Based on the equation, we can see that the denominator values 36 (under x) and 25 (under y) are the squares of the semi-major and semi-minor axes, respectively. Since 36 < 25, this means the ellipse is horizontally oriented.
b) Now, let's determine the coordinates:
The center = (1, -3) since the form of the equation is [(x - h)²/a² + (y - k)²/b² = 1, and (h, k) represents the center point.
Major vertices: To find the major vertices, we move a distance of ±a units from the center along the major axis. In this case, a = √36 = 6. So, the major vertices are (1 ± 6, -3), which simplifies to (-5, -3) and (7, -3).
Minor vertices: To find the minor vertices, we move a distance of ±b units from the center along the minor axis. In this case, b = √25 = 5. So, the minor vertices are (1, -3 - 5) and (1, -3 + 5), which simplifies to (1, -8) and (1, 2).
To find the foci, we can use the equation c² = a² - b². In this case, c² = 36 - 25 = 11. Taking the square root, c = √11. Since the ellipse is horizontally oriented, the foci will be at (1 ± √11, -3).
Therefore, for the given equations:
1) a) The ellipse is vertically oriented.
b) Center = (-4, 0)
Major vertices = (-4, 8), (-4, -8)
Minor vertices = (-14, 0), (6, 0)
Foci = (-4, 6), (-4, -6)
2) a) The ellipse is horizontally oriented.
b) Center = (1, -3)
Major vertices = (-5, -3), (7, -3)
Minor vertices = (1, -8), (1, 2)
Foci = (1 - √11, -3), (1 + √11, -3)