The angle between the tangents to the parabola `y^(2)=4ax` at the points where it intersects with the line x-y-a= 0 is

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. `(pi)/(2)`

Answer

To find the angle between the tangents to the parabola \( y^2 = 4ax \) at the points where it intersects with the line \( x - y - a = 0 \), we can follow these steps:

### Step 1: Find the points of intersection
We need to find the points where the parabola intersects the line. We can substitute \( y \) from the line equation into the parabola's equation.

The line equation can be rearranged as:
\[ y = x - a \]

Now, substituting this into the parabola's equation \( y^2 = 4ax \):
\[ (x - a)^2 = 4ax \]

Expanding this gives:
\[ x^2 - 2ax + a^2 = 4ax \]
\[ x^2 - 6ax + a^2 = 0 \]

### Step 2: Solve the quadratic equation
To find the \( x \)-coordinates of the intersection points, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -6a, c = a^2 \):
\[ x = \frac{6a \pm \sqrt{(-6a)^2 - 4 \cdot 1 \cdot a^2}}{2 \cdot 1} \]
\[ x = \frac{6a \pm \sqrt{36a^2 - 4a^2}}{2} \]
\[ x = \frac{6a \pm \sqrt{32a^2}}{2} \]
\[ x = \frac{6a \pm 4a\sqrt{2}}{2} \]
\[ x = 3a \pm 2a\sqrt{2} \]

Thus, the \( x \)-coordinates of the intersection points are \( 3a + 2a\sqrt{2} \) and \( 3a - 2a\sqrt{2} \).

### Step 3: Find the corresponding \( y \)-coordinates
Using the line equation \( y = x - a \):
1. For \( x = 3a + 2a\sqrt{2} \):
\[ y = (3a + 2a\sqrt{2}) - a = 2a + 2a\sqrt{2} = 2a(1 + \sqrt{2}) \]

2. For \( x = 3a - 2a\sqrt{2} \):
\[ y = (3a - 2a\sqrt{2}) - a = 2a - 2a\sqrt{2} = 2a(1 - \sqrt{2}) \]

Thus, the points of intersection are:
\[ (3a + 2a\sqrt{2}, 2a(1 + \sqrt{2})) \quad \text{and} \quad (3a - 2a\sqrt{2}, 2a(1 - \sqrt{2})) \]

### Step 4: Find the tangents at these points
The equation of the tangent to the parabola \( y^2 = 4ax \) at a point \( (x_1, y_1) \) is given by:
\[ yy_1 = 2a(x + x_1) \]

For the point \( (3a + 2a\sqrt{2}, 2a(1 + \sqrt{2})) \):
\[ y(2a(1 + \sqrt{2})) = 2a(x + (3a + 2a\sqrt{2})) \]
Simplifying gives the equation of the first tangent.

For the point \( (3a - 2a\sqrt{2}, 2a(1 - \sqrt{2})) \):
\[ y(2a(1 - \sqrt{2})) = 2a(x + (3a - 2a\sqrt{2})) \]
This gives the equation of the second tangent.

### Step 5: Determine the angle between the tangents
Since the line \( x - y - a = 0 \) is a focal chord of the parabola, the angle between the tangents at the points where the parabola intersects this line is \( \frac{\pi}{2} \) (90 degrees).

### Final Answer
Thus, the angle between the tangents to the parabola at the points of intersection with the line is:
\[ \frac{\pi}{2} \]