can someone help me get started

[mattflint50]

a satelite is moving in an elliptical orbit around a planet. The equation of its planar orbit is 3x^2+4y^2=2. If the velocity of the satelite in the y direction is 10 when the y coordinate of the satelite is 2 what is the velocity in the x direction at that time?

I was given this problem, been have never found a related rate for an ellipse. I cannot find an example anywhere. Please help me get started.

 

[Euler]

:!:

 

[soroban]

Hello, mattflint50!

I assume there is a typo in the problem . . .

A satellite is moving in an elliptical orbit around a planet.
The equation of its planar orbit is \(\displaystyle 3x^2\,+\,4y^2\:=\;2.\)
If the velocity of the satellite in the y direction is 10 when the y coordinate of the satellite is 2,
what is the velocity in the x direction at that time?

The equation is: . \(\displaystyle 3x^2\,+\,4y^2\:=\:2\;\;\Rightarrow\;\;\frac{3x^2}{2}\,+\,2y^2\:=\:1\;\;\Rightarrow\;\;\frac{x^2}{\frac{2}{3}}\,+\,\frac{y^2}{\frac{1}{2}}\:=\:1\)

The ellipse has a semimajor axis of: \(\displaystyle a\,=\,\sqrt{\frac{2}{3}}\,\approx\,0.816\;\;\Rightarrow\;\;-0.816\,\leq\,x\,\leq\,0.816\)
. . . and a semiminor axis of: \(\displaystyle b\,=\,\sqrt{\frac{1}{2}}\,\approx\,0.707\;\;\Rightarrow\;\;-0.707\,\leq\,y\,\leq\,0.707\)

Hence, the y coordinate of the satellite cannot be 2.