Question

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

• A. 80
• B. 72
• C. 96
• D. 75

Hint:

When solving problems involving trigonometric identities and equations, carefully match the coefficients of terms when simplifying the expression.

Answer 1

The Correct Option is D

Step 1: Parametric Equations Analysis

The points are defined by the following parametric equations:
\[ x = 3 \tan\left( \theta + \frac{\pi}{3} \right) \]
\[ y = 2 \tan\left( \theta + \frac{\pi}{6} \right) \]

Step 2: Application of Trigonometric Identities

Let the following substitutions be made:
\[ A = \theta + \frac{\pi}{3} \]
\[ B = \theta + \frac{\pi}{6} \]

This implies that \( A - B = \frac{\pi}{6} \). Utilizing the tangent of a difference identity:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

Given that \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \), we substitute the parametric expressions:
\[ \frac{1}{\sqrt{3}} = \frac{\frac{x}{3} - \frac{y}{2}}{1 + \frac{x}{3} \cdot \frac{y}{2}} \]

Step 3: Equation Simplification

Simplify the numerator and denominator of the right-hand side:
\[ \frac{1}{\sqrt{3}} = \frac{\frac{2x - 3y}{6}}{\frac{6 + xy}{6}} = \frac{2x - 3y}{6 + xy} \]

Multiply both sides by \(6 + xy\):
\[ \frac{6 + xy}{\sqrt{3}} = 2x - 3y \]

Multiply by \( \sqrt{3} \) to clear the denominator:
\[ 6 + xy = 2\sqrt{3}x - 3\sqrt{3}y \]

Rearrange the terms to match a standard form:
\[ xy - 2\sqrt{3}x + 3\sqrt{3}y + 6 = 0 \]

Step 4: Comparison with the General Curve Equation

The provided general equation for a curve is:
\[ xy + \alpha x + \beta y + \gamma = 0 \]

By comparing the coefficients of our derived equation with the general form, we find:
\[ \begin{align} \alpha &= -2\sqrt{3} \\ \beta &= 3\sqrt{3} \\ \gamma &= 6 \end{align} \]

Step 5: Calculation of \( \alpha^2 + \beta^2 + \gamma^2 \)

Calculate the square of each coefficient:
\[ \begin{align} \alpha^2 &= (-2\sqrt{3})^2 = 12 \\ \beta^2 &= (3\sqrt{3})^2 = 27 \\ \gamma^2 &= 6^2 = 36 \end{align} \]

Sum the squared values:
\[ \alpha^2 + \beta^2 + \gamma^2 = 12 + 27 + 36 = 75 \]

Step 6: Identification of the Correct Option

The computed value matches option (4).