Animated Solution for Mathematics - Trigonometry: If for $\theta \in [-\frac{\pi }{3},0]$, the points $(x,y)=(3\tan (\theta +\frac{\pi }{3}),2\tan (\theta +\frac{\pi }{6}))$ lie on $xy+\alpha x+\beta y+\gamma =0$, then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is equal to :

Welcome, future engineers! Today, we are going to embark on a journey through the elegant world of parametric equations. Imagine a point $(x,y)$ dancing on a plane, its position governed by a single parameter $\theta$.

Our mission is to uncover the hidden path—the locus—that this point traces. The problem provides the coordinates:

At first glance, this might look like a daunting task, but let us break it down with the precision of a master architect.

Look closely at the angles inside the tangent functions. Let $A=\theta +\frac{\pi }{3}$ and $B=\theta +\frac{\pi }{6}$.

The parameter $\theta$ is the variable that makes the point move, but notice what happens when we look at the difference between these two angles:

The $\theta$ cancels out entirely! This is the "Aha!" moment. We have discovered a constant relationship between the two angles, which is the key to eliminating the parameter.

Now that we have our angles, let us express the coordinates in terms of these angles:

We need to connect these to the constant difference we found. This is where our trigonometric toolkit comes in. We use the identity for the tangent of a difference:

Since we know $A-B=\frac{\pi }{6}$, we know that $\tan (A-B)=\tan \left(\frac{\pi }{6}\right)=\frac{1}{\sqrt{3}}$.

Now, let us substitute our expressions for $\tan A$ and $\tan B$ into the identity:

This looks like a lot of fractions, but do not be intimidated. Let us simplify the numerator and denominator:

When we divide these, the $6$ in the denominator cancels out beautifully, leaving us with:

Cross-multiplying gives us $\sqrt{3}(2x-3y)=6+xy$, which rearranges to:

This is the equation of our locus. By comparing this with the given form $xy+\alpha x+\beta y+\gamma =0$, we identify the coefficients:

The final step is to calculate ${\alpha }^2+{\beta }^2+{\gamma }^2$. Squaring these values gives:

The final answer is 75.