CALCULUS: Tangent on an elliptical orbit to reach a point?

Question

You are on a spaceship that is orbiting a planet.  You have ran out of fuel and the ship is therefore not navigable.  However, you have an anti-gravity that acts instantaneously upon the press of a button.  At which point on the elliptical orbit should you press the anti-gravity button in order to reach a nearby planet.

The elliptical orbit is:
x^2 / 4 + y^2 / 9 = 1

The nearby planet is located at the point:
(40, 3)

Questions:
1.) At which point on the elliptic orbit should you press the anti-gravity button (if you are traveling clockwise on the coordinate system)?
2.) Why is it a good idea to make the substitution of z = x/2?
(hint: quartic polynomial)

? · Accepted Answer

x²/4 + y²/9 = 1 , so 9x²+4y²=36
y'(x) = -(9x)/(4y)
4y(y-3) = -9x(x-40) the point-slope form of a line becomes y=-30x+3 which we substitute back into 9x²+4y²=36
9x²+4(3-30x)²=36 which simplifies to 3609x²-720x=0 which factorizes to 9x(401x-80)=0
At (0,3) we are on a tangent through (40,3), but we are moving in the opposite direction, so discard this solution.
The remaining solution is (80/401, -1197/401)

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CALCULUS: Tangent on an elliptical orbit to reach a point?

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