Range contact chord lengths when $\frac{x^2}{25}+\frac{y^2}{16}=1$ is touching a circle inside

Asked Jan 18 '18 at 11:23

Active Jan 24 '18 at 06:42

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A circle passes through a point on $x$ axis vertically ( like point $(3,0) $ shown) and has its center on $y=0$ and touches the ellipse at $A$ and $B$. What is the range of chord of contact $AB$ ?

Known answer: $$ 4 < AB <7 \,$$

My approach in the sketch

EDIT:

A variable circle centered on $x$ axis touches an ellipse $ (a=5,b=4),$ passes through a typical point such as $(3,0)$ as shown. Find range of distance between ellipse/circle tangent points ( length of contact chord ).

edited Jan 18 '18 at 20:45

Narasimham

40,495

asked Jan 18 '18 at 11:23

Piesquareisg

I have assumed the parametric coordinates of the ellipse, so than we to find maximum value of 8sin theta, can't figure out what to do next – Piesquareisg Jan 18 '18 at 12:26
Is it only touch or also intersection is acceptable? – Mostafa Ayaz Jan 18 '18 at 13:05
The circle only touches – Piesquareisg Jan 18 '18 at 13:10
I have placed a comment in your question area as the sketch given is too big for the comments area. If not ok, feel free to roll it back. – Narasimham Jan 18 '18 at 20:34
To find the range, $AB$ first set up equations of circle and ellipse. Contact occurs when discriminant vanishes while solving for intersection points. – Narasimham Jan 18 '18 at 20:42
Related to https://math.stackexchange.com/q/1972994/265466. – amd Jan 19 '18 at 03:58

Range contact chord lengths when $\frac{x^2}{25}+\frac{y^2}{16}=1$ is touching a circle inside

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