If it is not possible to draw any tangent from the point (1/4, 1) to the parabola `y^(2)=4xcostheta+sin^(2)theta`, then `theta` belongs to

Options

A. `[-pi/2,pi/2]`
B. `[-pi/2,pi/2]-{0}`
C. `(-pi/2,pi/2)-{0}`
D. none of these

Answer

To solve the problem, we need to determine the values of \(\theta\) for which it is not possible to draw any tangent from the point \((\frac{1}{4}, 1)\) to the parabola given by the equation:

\[ y^2 = 4x \cos \theta + \sin^2 \theta \]

### Step 1: Identify the point and the parabola
The point from which we want to draw tangents is \(P\left(\frac{1}{4}, 1\right)\) and the parabola is defined as \(y^2 = 4x \cos \theta + \sin^2 \theta\).

### Step 2: Use the condition for tangents
For a point to be able to draw tangents to a parabola, the discriminant of the quadratic equation formed must be non-negative. If it is negative, it means the point lies inside the parabola and no tangents can be drawn.

### Step 3: Substitute the point into the parabola equation
We substitute \(x = \frac{1}{4}\) and \(y = 1\) into the parabola equation:

\[ 1^2 = 4\left(\frac{1}{4}\right) \cos \theta + \sin^2 \theta \]

This simplifies to:

\[ 1 = \cos \theta + \sin^2 \theta \]

### Step 4: Rearranging the equation
Rearranging gives:

\[ \sin^2 \theta = 1 - \cos \theta \]

### Step 5: Use the Pythagorean identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can substitute \(\sin^2 \theta\):

\[ 1 - \cos^2 \theta = 1 - \cos \theta \]

This simplifies to:

\[ \cos^2 \theta - \cos \theta = 0 \]

### Step 6: Factor the equation
Factoring gives:

\[ \cos \theta (\cos \theta - 1) = 0 \]

### Step 7: Solve for \(\theta\)
This implies:

1. \(\cos \theta = 0\) or
2. \(\cos \theta - 1 = 0\) which means \(\cos \theta = 1\)

### Step 8: Determine the values of \(\theta\)
1. \(\cos \theta = 0\) gives \(\theta = \frac{\pi}{2} + n\pi\) for \(n \in \mathbb{Z}\).
2. \(\cos \theta = 1\) gives \(\theta = 2n\pi\) for \(n \in \mathbb{Z}\).

### Step 9: Determine the range of \(\theta\)
From the context of the problem, we need to find the range of \(\theta\) such that it is not possible to draw any tangents. This occurs when \(\cos \theta\) is between 0 and 1, which corresponds to:

\[ \theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]

### Final Answer
Thus, the values of \(\theta\) for which it is not possible to draw any tangent from the point \((\frac{1}{4}, 1)\) to the parabola are:

\[ \theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \]