circle inside an ellipse with fixed width but variable length
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Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r
geometry functions
asked Jan 10 at 10:53
nicknamenickname
31
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add a comment |
$begingroup$
Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r
geometry functions
asked Jan 10 at 10:53
nicknamenickname
31
$endgroup$
$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
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– David K
Jan 10 at 16:20
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Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16
add a comment |
$begingroup$
Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r
geometry functions
asked Jan 10 at 10:53
nicknamenickname
31
$endgroup$
Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r
geometry functions
geometry functions
asked Jan 10 at 10:53
nicknamenickname
31
asked Jan 10 at 10:53
nicknamenickname
31
edited Jan 10 at 10:55
nickname
asked Jan 10 at 10:53
nicknamenickname
31
asked Jan 10 at 10:53
nicknamenickname
31
asked Jan 10 at 10:53
nicknamenickname
31
31
$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
$endgroup$
– David K
Jan 10 at 16:20
$begingroup$
Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16
add a comment |
$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
$endgroup$
– David K
Jan 10 at 16:20
$begingroup$
Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16
$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
$endgroup$
– David K
Jan 10 at 16:20
$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
$endgroup$
– David K
Jan 10 at 16:20
$begingroup$
Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16
$begingroup$
Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16
add a comment |
1 Answer
1
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The smallest value of $b$ is that leading to tangency between ellipse and circle.
Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
The smallest value of $b$ is that leading to tangency between ellipse and circle.
Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
$endgroup$
add a comment |
$begingroup$
The smallest value of $b$ is that leading to tangency between ellipse and circle.
Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
$endgroup$
add a comment |
$begingroup$
The smallest value of $b$ is that leading to tangency between ellipse and circle.
Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
$endgroup$
The smallest value of $b$ is that leading to tangency between ellipse and circle.
Write down the equations of the ellipse and circle, eliminate $y^2$ to get a single quadratic equation in $x$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
answered Jan 10 at 14:14
AretinoAretino
25.8k31545
25.8k31545
add a comment |
add a comment |
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$begingroup$
If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$?
$endgroup$
– David K
Jan 10 at 16:20
$begingroup$
Hint: Show that the center of the circle lies on a focus of the ellipse.
$endgroup$
– amd
Jan 11 at 2:16