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 :

Imagine you are standing before a complex, shifting landscape. You have a point $(x,y)$ that is dancing across a plane, but its position is dictated by a hidden parameter, $\theta$.

This is the essence of parametric geometry—a beautiful, rhythmic dance where the variables $x$ and $y$ are not directly related, but are instead tethered to a common master, $\theta$. Our goal today is to strip away this mask and find the true, static path—the locus—that this point traces.

Analyzing the Coordinates

We begin with the given coordinates:

At first glance, this looks intimidating. We have two different tangent functions, each with a different phase shift.

There is a hidden symmetry here. The parameter $\theta$ is present in both, but the difference between these two angles is a constant. If we can isolate this difference, we can eliminate $\theta$ entirely.

Defining $A$ and $B$

Let us define $A=\theta +\frac{\pi }{3}$ and $B=\theta +\frac{\pi }{6}$. By doing this, we are not just renaming variables; we are simplifying our mental model.

Now, our coordinates become $x=3\tan A$ and $y=2\tan B$. We can easily isolate the trigonometric parts:

Now, consider the difference $A-B$: The $\theta$ has vanished! We have successfully decoupled the geometry from the parameter.

Applying the Identity

Now that we have $\tan A$ and $\tan B$, and we know the value of $A-B$, we need a bridge to connect them. That bridge is the compound angle identity:

We know that $\tan (A-B)=\tan (\frac{\pi }{6})=\frac{1}{\sqrt{3}}$. Substituting our expressions, we obtain:

Algebraic Manipulation

Now, we must be careful. Let us simplify the fraction by multiplying the numerator and denominator by $6$:

Cross-multiplying yields:

Rearranging everything to match the form $xy+\alpha x+\beta y+\gamma =0$, we arrive at:

By comparing this to the standard form, we identify our constants: $\alpha =-2\sqrt{3}$, $\beta =3\sqrt{3}$, and $\gamma =6$.

Calculating the Result

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

Summing these up:

And there it is! The answer is 75. Whenever you face a parametric problem in the future, remember this: look for the constant difference between the angles. It is the key that unlocks the door.