how is a standard equation derived form an ellipse?

Question

this is for my school project

Answer 1

The equation for an ellipse centered at the origin is:
x^2/a^2 + y^2/b^2 = 1
where
x = a cos M
y = b sin M

An ellipse is a strange character. At the short or minor-axis, in this case B, (0,b) you can draw a circle inside of it of radius B. and on the outside at the long or major-axis at A (a=0) you can draw a circle of radius A. Now this may be hard to visiualize with out a drawing, but imagine a line segment attached to the origin and starting at the + x-axis and moving toward the + y-axis then the angle formed between the segment and the x-axis is M. Then if you go to the points where the line segment intercepts the two circles then it is possilble to draw a right triangle with the right angle being at point P on the ellipse itself.
This is why that is important.
The easiest way to draw an ellipse is to know its two focus. Believe it or not a single one is called a focii, from latin. They are two points a distance from the origin. A circle is just an ellipse with both focus at the origin and later you will learn that parabolas are just elipses where the focus have past each other.
But here are two more important formulas and the reason why I have had to write so much.
BF =BF' =a
and
PF' + PF = 2a
All of this is important because a guy named Kepler said planets travel in ellipses and then he came up with this observation that makes up his 3rd Law which Newton used to find G and then give us gravity that is all based on the ellipse.

Answer 2

3. Solve the following system of equations, using the elimination method: 3x - 4y = -30 7x + 4y = 10 3x - 4y = -30 3x = -30 + 4y 3x + 30 = 4y 3/4x + 30/4 = y 7x + 4y = 10 4y = 10 - 7x y = 10/4 - 7/4x y-y = 0 (3/4x + 30/4) - (10/4 - 7/4x) = 0 3/4x + 30/4 - 10/4 + 7/4x = 0 10/4x + 20/4 = 0 10/4x = -20/4 10x = -20 x = -2 6. Solve the following system of equations, using the graphing method: 3x - 2y = 5 2x + 5y = -3 3x - 2y = 5 -2y = 5 - 3x y = -5/2 + 3/2x 2x + 5y = -3 5y = -3 - 2x y = -3/5 - 2/5x y = y -5/2 + 3/2x = -3/5 - 2/5x 3/2x + 2/5x = -3/5 + 5/2 15/10 x + 4/10x = -6/10 + 25/10 19/10x = 19/10 19x = 19 x = 1 10. Solve the following system of equations, using the substitution method: 4x - 3y = 33 x = - 4y - 25 4x - 3y = 33 -3y = 33 - 4x y = -33/3 + 4/3x y = -11 + 4/3x x = - 4y - 25 Substitute y by -11 + 4/3x x = -4 (-11 + 4/3x) - 25 x = 44 - 16/3x - 25 x + 16/3x = 19 3/3x + 16/3x = 19 19/3x = 19 1/3x = 1 x = 3

Answer 3

aX^2 + bY^2 = C (Constant)