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In your post, (a) is not an equation at all.
(e) is not an equation of an ellipse. It represents the single point (0,0), which is the origin of the coordinate system.
Your equation (g) looks suspicious.
I will do (c), (d) and (f) for you.
In order to understand my solution, you need to be familiar with the basic notions and formulas related to ellipses.
It is the NECESSARY condition to understand what is written in my post.
You can learn all the necessary introductory knowledge from my lessons
- Ellipse definition, canonical equation, characteristic points and elements
- General equation of an ellipse
- Identify elements of an ellipse given by its general equation
in this site.
I will assume that you did it.
(c) + = 1
It is the standard form of an ellipse equation.
Moreover, it is the canonical form of an ellipse equation.
The ellipse, described by this equation, has the center at the point (0,0), which is the origin of the coordinate system.
The major axis of this ellipse is VERTICAL axis "y", while the minor axis is HORIZONTAL axis "x".
This ellipse IS TALLER than WIDE.
The major semi-axis has the length of = units,
while the minor semi-axis has the length of = 3 units.
Therefore, the major axis has the length of units, while the minor axis has the length of 6 units.
(d) + = 32.
Divide both sides of this equation by 32. You will get an equivalent equation
+ = 1.
It is the standard form of an ellipse equation.
Moreover, it is the canonical form of an ellipse equation.
The ellipse, described by this equation, has the center at the point (0,0), which is the origin of the coordinate system.
The major axis of this ellipse is horizontal x-axis, while the minor axis is vertical y-axis.
The major semi-axis has the length of = units,
while the minor semi-axis has the length of = 2 units.
Therefore, the major axis has the length of units, while the minor axis has the length of 4 units.
(f) + = 1. <<<---=== Notice that your original equation was written INCORRECTLY, and I fixed it (!)
Your equation is
+ = 1.
It is the standard form of an ellipse equation.
( But not its canonical form ).
The ellipse, described by this equation, has the center at the point (5,-11) of the coordinate system.
The major axis of this ellipse is VERTICAL line parallel to y-axis, while the minor axis is the HORIZONTAL line parallel to x-axis.
This ellipse IS TALLER than WIDE.
The major semi-axis has the length of 12 units,
while the minor semi-axis has the length of 11 units.
Therefore, the major axis has the length of 24 units, while the minor axis has the length of 22 units.
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One advise for the future.
The rules of this forum require that every post goes with one problem.
Do not place many problem in your post - do not violate this rule.
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When you complete reading my post, do not forget to post your "THANKS" back to me.