Find angle between tangents drawn from origin to parabola `y^(2) =4a(x-a) `

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `(pi)/(3)`
D. `(pi)/(2)`

Answer

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

### Step 1: Identify the Parabola
The given parabola is \( y^2 = 4a(x - a) \). This is a standard form of a parabola that opens to the right.

### Step 2: Determine the Vertex and Directrix
From the equation \( y^2 = 4a(x - a) \):
- The vertex of the parabola is at the point \( (a, 0) \).
- The equation of the directrix can be derived from the standard form of the parabola. The directrix is given by \( x = a - \frac{a^2}{4a} = 0 \).

### Step 3: Understand the Properties of Tangents
For a parabola, the tangents drawn from any point on the directrix are perpendicular to each other. Since the origin (0, 0) lies on the directrix \( x = 0 \), we can conclude that the tangents drawn from the origin to the parabola will be perpendicular.

### Step 4: Calculate the Angle Between the Tangents
Since the tangents from the origin to the parabola are perpendicular, the angle \( \theta \) between them is:
\[ \theta = 90^\circ \text{ or } \frac{\pi}{2} \text{ radians} \]

### Final Answer
Thus, the angle between the tangents drawn from the origin to the parabola \( y^2 = 4a(x - a) \) is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.

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