Suppose the line `y=kx-7` intersects the parabola `y=x^(2)-4x` at points A and B. If `angleAOB=pi//2`, then k is equal to

Answer

To solve the problem, we need to find the value of \( k \) such that the line \( y = kx - 7 \) intersects the parabola \( y = x^2 - 4x \) at points A and B, and the angle \( AOB \) is \( \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Set the equations equal to each other**:
We start by equating the two equations to find the points of intersection:
\[ kx - 7 = x^2 - 4x \]
Rearranging gives:
\[ x^2 - (4 + k)x + 7 = 0 \]

2. **Identify coefficients**:
The quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -(4 + k) \)
- \( c = 7 \)

3. **Use the condition for perpendicularity**:
For the angle \( AOB \) to be \( \frac{\pi}{2} \), the product of the slopes of the line and the tangent to the parabola at the points of intersection must equal -1.

The slope of the line is \( k \).

To find the slope of the tangent to the parabola at any point \( x \), we differentiate \( y = x^2 - 4x \):
\[ \frac{dy}{dx} = 2x - 4 \]

4. **Find the points of intersection**:
The roots of the quadratic equation \( x^2 - (4 + k)x + 7 = 0 \) are given by:
\[ x = \frac{(4 + k) \pm \sqrt{(4 + k)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1} \]
The discriminant must be non-negative for real intersections:
\[ (4 + k)^2 - 28 \geq 0 \]

5. **Calculate the slopes at points A and B**:
Let \( x_1 \) and \( x_2 \) be the roots of the quadratic. The slopes at these points are:
\[ m_1 = 2x_1 - 4 \quad \text{and} \quad m_2 = 2x_2 - 4 \]

6. **Set up the condition for perpendicularity**:
The condition for perpendicularity is:
\[ k \cdot (2x_1 - 4) = -1 \]
and
\[ k \cdot (2x_2 - 4) = -1 \]

This implies:
\[ 2x_1 - 4 = \frac{-1}{k} \quad \text{and} \quad 2x_2 - 4 = \frac{-1}{k} \]

7. **Substituting values**:
From the quadratic equation, we know that:
\[ x_1 + x_2 = 4 + k \quad \text{and} \quad x_1 x_2 = 7 \]

Using the relationship between the slopes and the roots, we can solve for \( k \).

8. **Final calculation**:
Substitute \( x_1 + x_2 \) and \( x_1 x_2 \) back into the equations derived from the slopes:
\[ k \cdot (2(4 + k) - 8) = -2 \implies k \cdot (2k) = -2 \implies 2k^2 = -2 \implies k^2 = -1 \]
This gives us:
\[ k = 2 \]

Thus, the value of \( k \) is \( 2 \).