circle inside an ellipse with fixed width but variable length












0












$begingroup$


image of the diagram



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










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$endgroup$












  • $begingroup$
    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 $(-a,0)$?
    $endgroup$
    – 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
















0












$begingroup$


image of the diagram



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










share|cite|improve this question











$endgroup$












  • $begingroup$
    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 $(-a,0)$?
    $endgroup$
    – 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














0












0








0


1



$begingroup$


image of the diagram



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










share|cite|improve this question











$endgroup$




image of the diagram



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






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edited Jan 10 at 10:55







nickname

















asked Jan 10 at 10:53









nicknamenickname

1




1












  • $begingroup$
    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










  • $begingroup$
    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










1 Answer
1






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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.






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    active

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    1












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 10 at 14:14









        AretinoAretino

        23.5k21443




        23.5k21443






























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