Deriving the general equation of ellipses in cartesian form
0
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Given, the equation of a Cartesian circle is given in this general formula: $(x-a)^2 + (y-b)^2 = r^2$. This can be derived from the distance formula and Pythagoras's Theorem. However, how can I derive the general formula of an ellipse: $frac{(x-a)^2}{c} + frac{(y+b)^2}{d} = 1$ ?
algebra-precalculus geometry
edited Jan 18 at 8:30
KReiser
10.2k21435
asked Jan 18 at 7:11
Robin TingRobin Ting
94
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add a comment |
0
$begingroup$
Given, the equation of a Cartesian circle is given in this general formula: $(x-a)^2 + (y-b)^2 = r^2$. This can be derived from the distance formula and Pythagoras's Theorem. However, how can I derive the general formula of an ellipse: $frac{(x-a)^2}{c} + frac{(y+b)^2}{d} = 1$ ?
algebra-precalculus geometry
edited Jan 18 at 8:30
KReiser
10.2k21435
asked Jan 18 at 7:11
Robin TingRobin Ting
94
$endgroup$
$begingroup$
Take a circle and stretch one axis. The formula you have is not most general as you can also rotate an ellipse.
$endgroup$
– user121049
Jan 18 at 8:40
$begingroup$
perhaps it might be good to include what do you think an ellipse is.
$endgroup$
– Siong Thye Goh
Jan 18 at 9:50
$begingroup$
en.wikipedia.org/wiki/Ellipse#Equation
$endgroup$
– Aretino
Jan 18 at 10:46
1
$begingroup$
This depends on the definition of an ellipse you use. By the way your formula is not the general one. It describes only the ellipses with axes parallel to the coordinate ones.
$endgroup$
– user
Jan 18 at 13:03
add a comment |
0
0
0
$begingroup$
Given, the equation of a Cartesian circle is given in this general formula: $(x-a)^2 + (y-b)^2 = r^2$. This can be derived from the distance formula and Pythagoras's Theorem. However, how can I derive the general formula of an ellipse: $frac{(x-a)^2}{c} + frac{(y+b)^2}{d} = 1$ ?
algebra-precalculus geometry
edited Jan 18 at 8:30
KReiser
10.2k21435
asked Jan 18 at 7:11
Robin TingRobin Ting
94
$endgroup$
Given, the equation of a Cartesian circle is given in this general formula: $(x-a)^2 + (y-b)^2 = r^2$. This can be derived from the distance formula and Pythagoras's Theorem. However, how can I derive the general formula of an ellipse: $frac{(x-a)^2}{c} + frac{(y+b)^2}{d} = 1$ ?
algebra-precalculus geometry
algebra-precalculus geometry
edited Jan 18 at 8:30
KReiser
10.2k21435
asked Jan 18 at 7:11
Robin TingRobin Ting
94
edited Jan 18 at 8:30
KReiser
10.2k21435
asked Jan 18 at 7:11
Robin TingRobin Ting
94
edited Jan 18 at 8:30
KReiser
10.2k21435
edited Jan 18 at 8:30
KReiser
10.2k21435
edited Jan 18 at 8:30
KReiser
10.2k21435
10.2k21435
asked Jan 18 at 7:11
Robin TingRobin Ting
94
asked Jan 18 at 7:11
Robin TingRobin Ting
94
asked Jan 18 at 7:11
Robin TingRobin Ting
94
94
$begingroup$
Take a circle and stretch one axis. The formula you have is not most general as you can also rotate an ellipse.
$endgroup$
– user121049
Jan 18 at 8:40
$begingroup$
perhaps it might be good to include what do you think an ellipse is.
$endgroup$
– Siong Thye Goh
Jan 18 at 9:50
$begingroup$
en.wikipedia.org/wiki/Ellipse#Equation
$endgroup$
– Aretino
Jan 18 at 10:46
1
$begingroup$
This depends on the definition of an ellipse you use. By the way your formula is not the general one. It describes only the ellipses with axes parallel to the coordinate ones.
$endgroup$
– user
Jan 18 at 13:03
add a comment |
$begingroup$
Take a circle and stretch one axis. The formula you have is not most general as you can also rotate an ellipse.
$endgroup$
– user121049
Jan 18 at 8:40
$begingroup$
perhaps it might be good to include what do you think an ellipse is.
$endgroup$
– Siong Thye Goh
Jan 18 at 9:50
$begingroup$
en.wikipedia.org/wiki/Ellipse#Equation
$endgroup$
– Aretino
Jan 18 at 10:46
1
$begingroup$
This depends on the definition of an ellipse you use. By the way your formula is not the general one. It describes only the ellipses with axes parallel to the coordinate ones.
$endgroup$
– user
Jan 18 at 13:03
$begingroup$
Take a circle and stretch one axis. The formula you have is not most general as you can also rotate an ellipse.
$endgroup$
– user121049
Jan 18 at 8:40
$begingroup$
Take a circle and stretch one axis. The formula you have is not most general as you can also rotate an ellipse.
$endgroup$
– user121049
Jan 18 at 8:40
$begingroup$
perhaps it might be good to include what do you think an ellipse is.
$endgroup$
– Siong Thye Goh
Jan 18 at 9:50
$begingroup$
perhaps it might be good to include what do you think an ellipse is.
$endgroup$
– Siong Thye Goh
Jan 18 at 9:50
$begingroup$
en.wikipedia.org/wiki/Ellipse#Equation
$endgroup$
– Aretino
Jan 18 at 10:46
$begingroup$
en.wikipedia.org/wiki/Ellipse#Equation
$endgroup$
– Aretino
Jan 18 at 10:46
1
1
$begingroup$
This depends on the definition of an ellipse you use. By the way your formula is not the general one. It describes only the ellipses with axes parallel to the coordinate ones.
$endgroup$
– user
Jan 18 at 13:03
$begingroup$
This depends on the definition of an ellipse you use. By the way your formula is not the general one. It describes only the ellipses with axes parallel to the coordinate ones.
$endgroup$
– user
Jan 18 at 13:03
add a comment |
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$begingroup$
Take a circle and stretch one axis. The formula you have is not most general as you can also rotate an ellipse.
$endgroup$
– user121049
Jan 18 at 8:40
$begingroup$
perhaps it might be good to include what do you think an ellipse is.
$endgroup$
– Siong Thye Goh
Jan 18 at 9:50
$begingroup$
en.wikipedia.org/wiki/Ellipse#Equation
$endgroup$
– Aretino
Jan 18 at 10:46
1
$begingroup$
This depends on the definition of an ellipse you use. By the way your formula is not the general one. It describes only the ellipses with axes parallel to the coordinate ones.
$endgroup$
– user
Jan 18 at 13:03