Mixing
Understanding $lvert z-1 rvert+lvert z+1 rvert=7$ graphically
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$lvert z-1 rvert+lvert z+1 rvert=7$ is a circle of radius 3.5 if we use a computer algebra system to draw it. One can take $z:=x+iy$ and get the equation
$$sqrt(x-1)^2+y^2+sqrt(x+1)^2+y^2=7$$
Then we can square both sides and get another expression, but from that expression we still won't likely read the said circle.
So how can we actually understand, without using a computer algebra system, that $lvert z-1 rvert+lvert z+1 rvert=7$ represents the said circle? I'd guess this has something to do with the average equidistance of $z$ from the points $1$ and $-1$.
complex-analysis geometry
asked Jul 24 at 6:20
sequence
4,03711031
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up vote
2
down vote
favorite
$lvert z-1 rvert+lvert z+1 rvert=7$ is a circle of radius 3.5 if we use a computer algebra system to draw it. One can take $z:=x+iy$ and get the equation
$$sqrt(x-1)^2+y^2+sqrt(x+1)^2+y^2=7$$
Then we can square both sides and get another expression, but from that expression we still won't likely read the said circle.
So how can we actually understand, without using a computer algebra system, that $lvert z-1 rvert+lvert z+1 rvert=7$ represents the said circle? I'd guess this has something to do with the average equidistance of $z$ from the points $1$ and $-1$.
complex-analysis geometry
asked Jul 24 at 6:20
sequence
4,03711031
3
It's not a circle, but an ellipse. What makes you think it would be a circle?
– dxiv
Jul 24 at 6:21
2
According to your equation, the sum of distances from point $(x,y)$ to points (-1,0) and (1, 0) is constant (7). Therefore, this is ellipse, not a circle.
– Oldboy
Jul 24 at 6:26
1
"(W)without using a computer algebra system", two lines of computations lead to $$180x^2+196y^2=2205$$ which obviously describes an ellipse.
– Did
Jul 24 at 8:28
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
$lvert z-1 rvert+lvert z+1 rvert=7$ is a circle of radius 3.5 if we use a computer algebra system to draw it. One can take $z:=x+iy$ and get the equation
$$sqrt(x-1)^2+y^2+sqrt(x+1)^2+y^2=7$$
Then we can square both sides and get another expression, but from that expression we still won't likely read the said circle.
So how can we actually understand, without using a computer algebra system, that $lvert z-1 rvert+lvert z+1 rvert=7$ represents the said circle? I'd guess this has something to do with the average equidistance of $z$ from the points $1$ and $-1$.
complex-analysis geometry
asked Jul 24 at 6:20
sequence
4,03711031
$lvert z-1 rvert+lvert z+1 rvert=7$ is a circle of radius 3.5 if we use a computer algebra system to draw it. One can take $z:=x+iy$ and get the equation
$$sqrt(x-1)^2+y^2+sqrt(x+1)^2+y^2=7$$
Then we can square both sides and get another expression, but from that expression we still won't likely read the said circle.
So how can we actually understand, without using a computer algebra system, that $lvert z-1 rvert+lvert z+1 rvert=7$ represents the said circle? I'd guess this has something to do with the average equidistance of $z$ from the points $1$ and $-1$.
complex-analysis geometry
asked Jul 24 at 6:20
sequence
4,03711031
asked Jul 24 at 6:20
sequence
4,03711031
asked Jul 24 at 6:20
sequence
4,03711031
asked Jul 24 at 6:20
sequence
4,03711031
4,03711031
3
It's not a circle, but an ellipse. What makes you think it would be a circle?
– dxiv
Jul 24 at 6:21
2
According to your equation, the sum of distances from point $(x,y)$ to points (-1,0) and (1, 0) is constant (7). Therefore, this is ellipse, not a circle.
– Oldboy
Jul 24 at 6:26
1
"(W)without using a computer algebra system", two lines of computations lead to $$180x^2+196y^2=2205$$ which obviously describes an ellipse.
– Did
Jul 24 at 8:28
add a comment |Â
3
It's not a circle, but an ellipse. What makes you think it would be a circle?
– dxiv
Jul 24 at 6:21
2
According to your equation, the sum of distances from point $(x,y)$ to points (-1,0) and (1, 0) is constant (7). Therefore, this is ellipse, not a circle.
– Oldboy
Jul 24 at 6:26
1
"(W)without using a computer algebra system", two lines of computations lead to $$180x^2+196y^2=2205$$ which obviously describes an ellipse.
– Did
Jul 24 at 8:28
3
3
It's not a circle, but an ellipse. What makes you think it would be a circle?
– dxiv
Jul 24 at 6:21
It's not a circle, but an ellipse. What makes you think it would be a circle?
– dxiv
Jul 24 at 6:21
2
2
According to your equation, the sum of distances from point $(x,y)$ to points (-1,0) and (1, 0) is constant (7). Therefore, this is ellipse, not a circle.
– Oldboy
Jul 24 at 6:26
According to your equation, the sum of distances from point $(x,y)$ to points (-1,0) and (1, 0) is constant (7). Therefore, this is ellipse, not a circle.
– Oldboy
Jul 24 at 6:26
1
1
"(W)without using a computer algebra system", two lines of computations lead to $$180x^2+196y^2=2205$$ which obviously describes an ellipse.
– Did
Jul 24 at 8:28
"(W)without using a computer algebra system", two lines of computations lead to $$180x^2+196y^2=2205$$ which obviously describes an ellipse.
– Did
Jul 24 at 8:28
add a comment |Â
2 Answers
2
active
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4
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accepted
You're looking at the locus of points which have the property that the sum of their distances from $-1$ and $1$ is $7$. This is the locus definition of an ellipse.
answered Jul 24 at 6:26
Theo Bendit
12k1843
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up vote
2
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Although your example is mistaken, I still think the general question is useful.
One observation is that $|z-z_0|$, for any constant complex value $z_0$, represents the distance of $z$ from $z_0$. For instance $|z-1|$ represents the distance of a complex number (in the complex plane) from $1$. So the equation
$$
|z-1| = 3
$$
is the equation of a circle (again, in the complex plane) of radius $3$ and centered on the complex value $1$. Once one internalizes this sort of thing (and knows the definition of an ellipse as the locus of points with the distance sum property), the equation
$$
|z-1|+|z+1| = 7
$$
can be read off quite straightforwardly as an ellipse with foci at $-1$ and $1$.
answered Jul 24 at 6:51
Brian Tung
25.2k32453
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
You're looking at the locus of points which have the property that the sum of their distances from $-1$ and $1$ is $7$. This is the locus definition of an ellipse.
answered Jul 24 at 6:26
Theo Bendit
12k1843
add a comment |Â
up vote
4
down vote
accepted
You're looking at the locus of points which have the property that the sum of their distances from $-1$ and $1$ is $7$. This is the locus definition of an ellipse.
answered Jul 24 at 6:26
Theo Bendit
12k1843
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
You're looking at the locus of points which have the property that the sum of their distances from $-1$ and $1$ is $7$. This is the locus definition of an ellipse.
answered Jul 24 at 6:26
Theo Bendit
12k1843
You're looking at the locus of points which have the property that the sum of their distances from $-1$ and $1$ is $7$. This is the locus definition of an ellipse.
answered Jul 24 at 6:26
Theo Bendit
12k1843
answered Jul 24 at 6:26
Theo Bendit
12k1843
answered Jul 24 at 6:26
Theo Bendit
12k1843
answered Jul 24 at 6:26
Theo Bendit
12k1843
12k1843
add a comment |Â
add a comment |Â
up vote
2
down vote
Although your example is mistaken, I still think the general question is useful.
One observation is that $|z-z_0|$, for any constant complex value $z_0$, represents the distance of $z$ from $z_0$. For instance $|z-1|$ represents the distance of a complex number (in the complex plane) from $1$. So the equation
$$
|z-1| = 3
$$
is the equation of a circle (again, in the complex plane) of radius $3$ and centered on the complex value $1$. Once one internalizes this sort of thing (and knows the definition of an ellipse as the locus of points with the distance sum property), the equation
$$
|z-1|+|z+1| = 7
$$
can be read off quite straightforwardly as an ellipse with foci at $-1$ and $1$.
answered Jul 24 at 6:51
Brian Tung
25.2k32453
add a comment |Â
up vote
2
down vote
Although your example is mistaken, I still think the general question is useful.
One observation is that $|z-z_0|$, for any constant complex value $z_0$, represents the distance of $z$ from $z_0$. For instance $|z-1|$ represents the distance of a complex number (in the complex plane) from $1$. So the equation
$$
|z-1| = 3
$$
is the equation of a circle (again, in the complex plane) of radius $3$ and centered on the complex value $1$. Once one internalizes this sort of thing (and knows the definition of an ellipse as the locus of points with the distance sum property), the equation
$$
|z-1|+|z+1| = 7
$$
can be read off quite straightforwardly as an ellipse with foci at $-1$ and $1$.
answered Jul 24 at 6:51
Brian Tung
25.2k32453
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Although your example is mistaken, I still think the general question is useful.
One observation is that $|z-z_0|$, for any constant complex value $z_0$, represents the distance of $z$ from $z_0$. For instance $|z-1|$ represents the distance of a complex number (in the complex plane) from $1$. So the equation
$$
|z-1| = 3
$$
is the equation of a circle (again, in the complex plane) of radius $3$ and centered on the complex value $1$. Once one internalizes this sort of thing (and knows the definition of an ellipse as the locus of points with the distance sum property), the equation
$$
|z-1|+|z+1| = 7
$$
can be read off quite straightforwardly as an ellipse with foci at $-1$ and $1$.
answered Jul 24 at 6:51
Brian Tung
25.2k32453
Although your example is mistaken, I still think the general question is useful.
One observation is that $|z-z_0|$, for any constant complex value $z_0$, represents the distance of $z$ from $z_0$. For instance $|z-1|$ represents the distance of a complex number (in the complex plane) from $1$. So the equation
$$
|z-1| = 3
$$
is the equation of a circle (again, in the complex plane) of radius $3$ and centered on the complex value $1$. Once one internalizes this sort of thing (and knows the definition of an ellipse as the locus of points with the distance sum property), the equation
$$
|z-1|+|z+1| = 7
$$
can be read off quite straightforwardly as an ellipse with foci at $-1$ and $1$.
answered Jul 24 at 6:51
Brian Tung
25.2k32453
answered Jul 24 at 6:51
Brian Tung
25.2k32453
answered Jul 24 at 6:51
Brian Tung
25.2k32453
answered Jul 24 at 6:51
Brian Tung
25.2k32453
25.2k32453
add a comment |Â
add a comment |Â
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3
It's not a circle, but an ellipse. What makes you think it would be a circle?
– dxiv
Jul 24 at 6:21
2
According to your equation, the sum of distances from point $(x,y)$ to points (-1,0) and (1, 0) is constant (7). Therefore, this is ellipse, not a circle.
– Oldboy
Jul 24 at 6:26
1
"(W)without using a computer algebra system", two lines of computations lead to $$180x^2+196y^2=2205$$ which obviously describes an ellipse.
– Did
Jul 24 at 8:28