Geometric or analytic proof that in hyperbola, $c^2=a^2+b^2$

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How to prove (geometric or analytic) that in hyperbola $c^2=a^2+b^2$? Given that $a$ is the undirected distance of the center to one of the vertices, $b$ is the undirected distance of one of the endpoints of the conjugate axis and $c$ is the undirected distance from the center to one of the foci.




conic-sections




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  • This should be tagged as conic-sections, geometry instead of algebraic-geometry.
    – user_of_math
    Oct 6 '14 at 4:40










  • Are $a$ and $b$ simply semi major and minor axes?
    – Kaster
    Oct 6 '14 at 4:43















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How to prove (geometric or analytic) that in hyperbola $c^2=a^2+b^2$? Given that $a$ is the undirected distance of the center to one of the vertices, $b$ is the undirected distance of one of the endpoints of the conjugate axis and $c$ is the undirected distance from the center to one of the foci.




conic-sections




share|cite|improve this question






















  • This should be tagged as conic-sections, geometry instead of algebraic-geometry.
    – user_of_math
    Oct 6 '14 at 4:40










  • Are $a$ and $b$ simply semi major and minor axes?
    – Kaster
    Oct 6 '14 at 4:43













up vote
0
down vote

favorite









up vote
0
down vote

favorite











How to prove (geometric or analytic) that in hyperbola $c^2=a^2+b^2$? Given that $a$ is the undirected distance of the center to one of the vertices, $b$ is the undirected distance of one of the endpoints of the conjugate axis and $c$ is the undirected distance from the center to one of the foci.




conic-sections




share|cite|improve this question














How to prove (geometric or analytic) that in hyperbola $c^2=a^2+b^2$? Given that $a$ is the undirected distance of the center to one of the vertices, $b$ is the undirected distance of one of the endpoints of the conjugate axis and $c$ is the undirected distance from the center to one of the foci.





conic-sections





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edited Oct 6 '14 at 15:27

























asked Oct 6 '14 at 4:32









MKA

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  • This should be tagged as conic-sections, geometry instead of algebraic-geometry.
    – user_of_math
    Oct 6 '14 at 4:40










  • Are $a$ and $b$ simply semi major and minor axes?
    – Kaster
    Oct 6 '14 at 4:43

















  • This should be tagged as conic-sections, geometry instead of algebraic-geometry.
    – user_of_math
    Oct 6 '14 at 4:40










  • Are $a$ and $b$ simply semi major and minor axes?
    – Kaster
    Oct 6 '14 at 4:43
















This should be tagged as conic-sections, geometry instead of algebraic-geometry.
– user_of_math
Oct 6 '14 at 4:40




This should be tagged as conic-sections, geometry instead of algebraic-geometry.
– user_of_math
Oct 6 '14 at 4:40












Are $a$ and $b$ simply semi major and minor axes?
– Kaster
Oct 6 '14 at 4:43





Are $a$ and $b$ simply semi major and minor axes?
– Kaster
Oct 6 '14 at 4:43











1 Answer
1






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Note that the eccentricity of a hyperbola is defined as $e = sqrt1+fracb^2a^2$. You know that $c = ae$ (by definition), therefore $c^2 = a^2 e^2 = a^2 (1+ fracb^2a^2) = a^2 + b^2$.






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  • How does the eccentricity of a hyperbola is defined as $e=√1+b2/a2$ ?
    – MKA
    Oct 6 '14 at 15:26











  • That and the linear eccentricity $c$ are defined/explained here: en.wikipedia.org/wiki/Conic_section#Features
    – James Harrison
    Oct 6 '14 at 22:06










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1 Answer
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active

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Note that the eccentricity of a hyperbola is defined as $e = sqrt1+fracb^2a^2$. You know that $c = ae$ (by definition), therefore $c^2 = a^2 e^2 = a^2 (1+ fracb^2a^2) = a^2 + b^2$.






share|cite|improve this answer




















  • How does the eccentricity of a hyperbola is defined as $e=√1+b2/a2$ ?
    – MKA
    Oct 6 '14 at 15:26











  • That and the linear eccentricity $c$ are defined/explained here: en.wikipedia.org/wiki/Conic_section#Features
    – James Harrison
    Oct 6 '14 at 22:06














up vote
0
down vote













Note that the eccentricity of a hyperbola is defined as $e = sqrt1+fracb^2a^2$. You know that $c = ae$ (by definition), therefore $c^2 = a^2 e^2 = a^2 (1+ fracb^2a^2) = a^2 + b^2$.






share|cite|improve this answer




















  • How does the eccentricity of a hyperbola is defined as $e=√1+b2/a2$ ?
    – MKA
    Oct 6 '14 at 15:26











  • That and the linear eccentricity $c$ are defined/explained here: en.wikipedia.org/wiki/Conic_section#Features
    – James Harrison
    Oct 6 '14 at 22:06












up vote
0
down vote










up vote
0
down vote









Note that the eccentricity of a hyperbola is defined as $e = sqrt1+fracb^2a^2$. You know that $c = ae$ (by definition), therefore $c^2 = a^2 e^2 = a^2 (1+ fracb^2a^2) = a^2 + b^2$.






share|cite|improve this answer












Note that the eccentricity of a hyperbola is defined as $e = sqrt1+fracb^2a^2$. You know that $c = ae$ (by definition), therefore $c^2 = a^2 e^2 = a^2 (1+ fracb^2a^2) = a^2 + b^2$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Oct 6 '14 at 4:55









James Harrison

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  • How does the eccentricity of a hyperbola is defined as $e=√1+b2/a2$ ?
    – MKA
    Oct 6 '14 at 15:26











  • That and the linear eccentricity $c$ are defined/explained here: en.wikipedia.org/wiki/Conic_section#Features
    – James Harrison
    Oct 6 '14 at 22:06
















  • How does the eccentricity of a hyperbola is defined as $e=√1+b2/a2$ ?
    – MKA
    Oct 6 '14 at 15:26











  • That and the linear eccentricity $c$ are defined/explained here: en.wikipedia.org/wiki/Conic_section#Features
    – James Harrison
    Oct 6 '14 at 22:06















How does the eccentricity of a hyperbola is defined as $e=√1+b2/a2$ ?
– MKA
Oct 6 '14 at 15:26





How does the eccentricity of a hyperbola is defined as $e=√1+b2/a2$ ?
– MKA
Oct 6 '14 at 15:26













That and the linear eccentricity $c$ are defined/explained here: en.wikipedia.org/wiki/Conic_section#Features
– James Harrison
Oct 6 '14 at 22:06




That and the linear eccentricity $c$ are defined/explained here: en.wikipedia.org/wiki/Conic_section#Features
– James Harrison
Oct 6 '14 at 22:06

















 

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