Angle between the tangents drawn to parabola `y^(2)+4a^(2)-4ax=0`, from origin is :

Options

A. `30^(@)`
B. `tan^(-1)(2)`
C. `90^(@)`
D. `tan^(-1)((1)/(2))`

Explanation

To find the angle between the tangents drawn to the parabola \( y^2 + 4a^2 - 4ax = 0 \) from the origin, we can follow these steps:

### Step 1: Rewrite the equation of the parabola
The given equation of the parabola is:
\[ y^2 + 4a^2 - 4ax = 0 \]
We can rearrange this to:
\[ y^2 = 4ax - 4a^2 \]

### Step 2: Substitute the tangent line equation
The equation of a line passing through the origin can be expressed as:
\[ y = mx \]
where \( m \) is the slope of the line. We will substitute \( y = mx \) into the parabola's equation.

### Step 3: Substitute and simplify
Substituting \( y = mx \) into the parabola's equation gives:
\[ (mx)^2 + 4a^2 - 4ax = 0 \]
This simplifies to:
\[ m^2x^2 - 4ax + 4a^2 = 0 \]

### Step 4: Apply the condition for tangency
For the line to be tangent to the parabola, the discriminant of this quadratic equation must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = m^2 \), \( b = -4a \), and \( c = 4a^2 \). Therefore, we have:
\[ D = (-4a)^2 - 4(m^2)(4a^2) = 16a^2 - 16a^2m^2 \]
Setting the discriminant to zero for tangency:
\[ 16a^2(1 - m^2) = 0 \]

### Step 5: Solve for \( m \)
Since \( 16a^2 \) cannot be zero (as \( a \) is a constant), we have:
\[ 1 - m^2 = 0 \implies m^2 = 1 \implies m = \pm 1 \]

### Step 6: Write the equations of the tangents
The slopes of the tangents are \( m = 1 \) and \( m = -1 \). Therefore, the equations of the tangents are:
\[ y = x \quad \text{and} \quad y = -x \]

### Step 7: Find the angle between the tangents
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) can be calculated using the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Here, \( m_1 = 1 \) and \( m_2 = -1 \):
\[ \tan \theta = \left| \frac{1 - (-1)}{1 + (1)(-1)} \right| = \left| \frac{2}{0} \right| \]
Since the denominator is zero, this indicates that the angle \( \theta \) is \( 90^\circ \).

### Conclusion
The angle between the tangents drawn to the parabola from the origin is:
\[ \theta = 90^\circ \]