Where does a chord of an Ellipse equal to the length of the minor axis but running parallel to the major axis...
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Given an Ellipse, I need to know where a chord equal to the length of the minor axis but running parallel to the major axis cross the minor axis.
geometry conic-sections
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
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add a comment |
$begingroup$
Given an Ellipse, I need to know where a chord equal to the length of the minor axis but running parallel to the major axis cross the minor axis.
geometry conic-sections
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
$endgroup$
add a comment |
$begingroup$
Given an Ellipse, I need to know where a chord equal to the length of the minor axis but running parallel to the major axis cross the minor axis.
geometry conic-sections
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
$endgroup$
Given an Ellipse, I need to know where a chord equal to the length of the minor axis but running parallel to the major axis cross the minor axis.
geometry conic-sections
geometry conic-sections
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
asked Dec 12 '18 at 1:25
Jim DalvicJim Dalvic
11
11
add a comment |
add a comment |
1 Answer
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$begingroup$
$frac {x^2}{a^2} + frac {y^2}{b^2} = 1$
Let $a>b$
the minor axis has length $2b$
We have a chord with $x$ coordinates $-b,b$ and we need to find the y coordinates.
$frac {b^2}{a^2} + frac {y^2}{b^2} = 1\
frac {y^2}{b^2} = 1 - frac {b^2}{a^2}\
y^2 = b^2(frac {a^2 - b^2}{a^2})\
y = pmfrac {b}{a}sqrt {a^2 - b^2}$
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
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$begingroup$
Thank you. This seems to work well for any hand drawn examples that I tried. Much appreciated.
$endgroup$
– Jim Dalvic
Dec 12 '18 at 16:23
$begingroup$
I have a follow up question. It seems to me that for a subset of ellipses, (Eccentricity between about .099 and .37 or so) that once you find the point on the minor axis where the above chord passes through the minor axis, that all chords passing through that same point on the minor axis have the same length. Is this mathematically verifiable? This seems counter intuitive to me and may just be a result of my ellipses being a bit sloppy when I draw them. Any help on this would be appreciated.
$endgroup$
– Jim Dalvic
11 hours ago
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
$frac {x^2}{a^2} + frac {y^2}{b^2} = 1$
Let $a>b$
the minor axis has length $2b$
We have a chord with $x$ coordinates $-b,b$ and we need to find the y coordinates.
$frac {b^2}{a^2} + frac {y^2}{b^2} = 1\
frac {y^2}{b^2} = 1 - frac {b^2}{a^2}\
y^2 = b^2(frac {a^2 - b^2}{a^2})\
y = pmfrac {b}{a}sqrt {a^2 - b^2}$
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
$endgroup$
$begingroup$
Thank you. This seems to work well for any hand drawn examples that I tried. Much appreciated.
$endgroup$
– Jim Dalvic
Dec 12 '18 at 16:23
$begingroup$
I have a follow up question. It seems to me that for a subset of ellipses, (Eccentricity between about .099 and .37 or so) that once you find the point on the minor axis where the above chord passes through the minor axis, that all chords passing through that same point on the minor axis have the same length. Is this mathematically verifiable? This seems counter intuitive to me and may just be a result of my ellipses being a bit sloppy when I draw them. Any help on this would be appreciated.
$endgroup$
– Jim Dalvic
11 hours ago
add a comment |
$begingroup$
$frac {x^2}{a^2} + frac {y^2}{b^2} = 1$
Let $a>b$
the minor axis has length $2b$
We have a chord with $x$ coordinates $-b,b$ and we need to find the y coordinates.
$frac {b^2}{a^2} + frac {y^2}{b^2} = 1\
frac {y^2}{b^2} = 1 - frac {b^2}{a^2}\
y^2 = b^2(frac {a^2 - b^2}{a^2})\
y = pmfrac {b}{a}sqrt {a^2 - b^2}$
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
$endgroup$
$begingroup$
Thank you. This seems to work well for any hand drawn examples that I tried. Much appreciated.
$endgroup$
– Jim Dalvic
Dec 12 '18 at 16:23
$begingroup$
I have a follow up question. It seems to me that for a subset of ellipses, (Eccentricity between about .099 and .37 or so) that once you find the point on the minor axis where the above chord passes through the minor axis, that all chords passing through that same point on the minor axis have the same length. Is this mathematically verifiable? This seems counter intuitive to me and may just be a result of my ellipses being a bit sloppy when I draw them. Any help on this would be appreciated.
$endgroup$
– Jim Dalvic
11 hours ago
add a comment |
$begingroup$
$frac {x^2}{a^2} + frac {y^2}{b^2} = 1$
Let $a>b$
the minor axis has length $2b$
We have a chord with $x$ coordinates $-b,b$ and we need to find the y coordinates.
$frac {b^2}{a^2} + frac {y^2}{b^2} = 1\
frac {y^2}{b^2} = 1 - frac {b^2}{a^2}\
y^2 = b^2(frac {a^2 - b^2}{a^2})\
y = pmfrac {b}{a}sqrt {a^2 - b^2}$
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
$endgroup$
$frac {x^2}{a^2} + frac {y^2}{b^2} = 1$
Let $a>b$
the minor axis has length $2b$
We have a chord with $x$ coordinates $-b,b$ and we need to find the y coordinates.
$frac {b^2}{a^2} + frac {y^2}{b^2} = 1\
frac {y^2}{b^2} = 1 - frac {b^2}{a^2}\
y^2 = b^2(frac {a^2 - b^2}{a^2})\
y = pmfrac {b}{a}sqrt {a^2 - b^2}$
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
answered Dec 12 '18 at 1:33
Doug MDoug M
44.7k31854
44.7k31854
$begingroup$
Thank you. This seems to work well for any hand drawn examples that I tried. Much appreciated.
$endgroup$
– Jim Dalvic
Dec 12 '18 at 16:23
$begingroup$
I have a follow up question. It seems to me that for a subset of ellipses, (Eccentricity between about .099 and .37 or so) that once you find the point on the minor axis where the above chord passes through the minor axis, that all chords passing through that same point on the minor axis have the same length. Is this mathematically verifiable? This seems counter intuitive to me and may just be a result of my ellipses being a bit sloppy when I draw them. Any help on this would be appreciated.
$endgroup$
– Jim Dalvic
11 hours ago
add a comment |
$begingroup$
Thank you. This seems to work well for any hand drawn examples that I tried. Much appreciated.
$endgroup$
– Jim Dalvic
Dec 12 '18 at 16:23
$begingroup$
I have a follow up question. It seems to me that for a subset of ellipses, (Eccentricity between about .099 and .37 or so) that once you find the point on the minor axis where the above chord passes through the minor axis, that all chords passing through that same point on the minor axis have the same length. Is this mathematically verifiable? This seems counter intuitive to me and may just be a result of my ellipses being a bit sloppy when I draw them. Any help on this would be appreciated.
$endgroup$
– Jim Dalvic
11 hours ago
$begingroup$
Thank you. This seems to work well for any hand drawn examples that I tried. Much appreciated.
$endgroup$
– Jim Dalvic
Dec 12 '18 at 16:23
$begingroup$
Thank you. This seems to work well for any hand drawn examples that I tried. Much appreciated.
$endgroup$
– Jim Dalvic
Dec 12 '18 at 16:23
$begingroup$
I have a follow up question. It seems to me that for a subset of ellipses, (Eccentricity between about .099 and .37 or so) that once you find the point on the minor axis where the above chord passes through the minor axis, that all chords passing through that same point on the minor axis have the same length. Is this mathematically verifiable? This seems counter intuitive to me and may just be a result of my ellipses being a bit sloppy when I draw them. Any help on this would be appreciated.
$endgroup$
– Jim Dalvic
11 hours ago
$begingroup$
I have a follow up question. It seems to me that for a subset of ellipses, (Eccentricity between about .099 and .37 or so) that once you find the point on the minor axis where the above chord passes through the minor axis, that all chords passing through that same point on the minor axis have the same length. Is this mathematically verifiable? This seems counter intuitive to me and may just be a result of my ellipses being a bit sloppy when I draw them. Any help on this would be appreciated.
$endgroup$
– Jim Dalvic
11 hours ago
add a comment |
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