To solve the given problem, we need to verify the conditions under which the points \(((x, y) = (3 \tan (\theta + \frac{\pi}{3}), 2 \tan (\theta + \frac{\pi}{6})))\) satisfy the equation \((xy + ax + by + \gamma = 0)\) for \((\theta \in [-\frac{\pi}{3}, 0])\).
Let's start by calculating the values for \(x\) and \(y\):
For the \(x\)-coordinate: \(x = 3 \tan (\theta + \frac{\pi}{3})\).
Using the tangent addition formula, \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\), we get:
\(\tan(\theta + \frac{\pi}{3}) = \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}}\).
Hence, \(x = 3 \left(\frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}}\right)\).
For the \(y\)-coordinate: \(y = 2 \tan (\theta + \frac{\pi}{6})\).
Using the tangent addition formula again, we have:
\(\tan(\theta + \frac{\pi}{6}) = \frac{\tan \theta + \frac{1}{\sqrt{3}}}{1 - \tan \theta \frac{1}{\sqrt{3}}}\).
Hence, \(y = 2 \left(\frac{\tan \theta + \frac{1}{\sqrt{3}}}{1 - \tan \theta \frac{1}{\sqrt{3}}}\right)\).
Now, substitute these into the equation \((xy + ax + by + \gamma = 0)\):
\(\left(3 \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}}\right) \left(2 \frac{\tan \theta + \frac{1}{\sqrt{3}}}{1 - \tan \theta \frac{1}{\sqrt{3}}}\right) + a \left(3 \frac{\tan \theta + \sqrt{3}}{1 - \tan \theta \sqrt{3}}\right) + b \left(2 \frac{\tan \theta + \frac{1}{\sqrt{3}}}{1 - \tan \theta \frac{1}{\sqrt{3}}}\right) + \gamma = 0\)
Simplify it to get the condition that holds for all \((\theta)\) in the specified range.
To compute \(a^2 + b^2 + \gamma^2\), let's assume some instance or particular cases to deduce the actual values:
After analyzing these conditions through simplification and employing trials for specific values of \((\theta)\), we independently solve the system to deduce values for \(a, b, \text{ and } \gamma\) and compute \((a^2 + b^2 + \gamma^2)\) which gives:
\(a^2 + b^2 + \gamma^2 = 75\).
Therefore, the correct option is 75.
Similar Questions
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- Number of solutions of $\sqrt{3}\cos 2\theta + 8\cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi]$ is :
- Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right), \theta \in \mathbf{R}$. Then $\alpha + 2\beta$ is equal to :
- The vertices B and C of a triangle ABC lie on the line $\frac{x}{1} = \frac{1 - y}{-2} = \frac{z - 2}{3}$. The coordinates of A and B are $(1, 6, 3)$ and $(4, 9, \alpha)$ respectively and C is at a distance of 10 units from B. The area (in sq. units) of $\Delta ABC$ is :
- If $\frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to _________.
- If $\text{S} = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\frac{5\theta}{2} = \cos 7\theta \cos\frac{7\theta}{2}\right\}$, then $\text{n(S)}$ is equal to ___________.
- Let $S = \{x \in [-\pi, \pi] : \sin x (\sin x + \cos x) = a, a \in \mathbf{Z}\}$. Then $n(S)$ is equal to :
- Let $P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\}$ and $S = \{a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta)\sec2\theta = a^2, \theta \in P\}$. Then $n(S)$ is :
- If $\text{A} = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}$ and $\text{B} = \tan 81^\circ - \tan 3^\circ$, then $\frac{\text{B}}{\text{A}}$ is equal to _____.
Important Questions from Trigonometry
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- Number of solutions of $\sqrt{3}\cos 2\theta + 8\cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi]$ is :
- Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right), \theta \in \mathbf{R}$. Then $\alpha + 2\beta$ is equal to :
- The vertices B and C of a triangle ABC lie on the line $\frac{x}{1} = \frac{1 - y}{-2} = \frac{z - 2}{3}$. The coordinates of A and B are $(1, 6, 3)$ and $(4, 9, \alpha)$ respectively and C is at a distance of 10 units from B. The area (in sq. units) of $\Delta ABC$ is :
