Mechanical Kinematics (Robotic Linkages) | Tony's Intertwingled Memex

A mechanical automation engineer is programming an industrial CNC plasma cutter on a manufacturing assembly line. The cutter head is constrained to move along an implicit elliptical track described by the equation:

4x^2 + 9y^2 = 36

where

x

and

y

represent coordinate positions in meters. During operation, optical telemetry confirms that the cutting head has reached the coordinate position:

(x, y) = \left(\frac{3}{\sqrt{2}}, \sqrt{2}\right)

Differentiate the implicit tracking equation with respect to $x$ using the Chain Rule to find an expression for the instantaneous tangent slope $\frac{dy}{dx}$ .
Evaluate $\frac{dy}{dx}$ at the operational position $\left(\frac{3}{\sqrt{2}}, \sqrt{2}\right)$ .
Determine the required orientation angle and normal direction for the plasma cutting nozzle so that it cuts completely perpendicular to the elliptical track.