The exhaustive set of values of k for which tangents drawn from the point (k + 3, k) to the parabola `y^(2)=4x`, are real, is

Options

A. (-2, 6)
B. `(-oo,-2)cup(6,oo)`
C. `(-oo,-2]cup[6,oo)`
D. None of these

Answer

To solve the problem of finding the exhaustive set of values of \( k \) for which tangents drawn from the point \( (k + 3, k) \) to the parabola \( y^2 = 4x \) are real, we will follow these steps:

### Step 1: Set up the equation for tangents
The equation of the parabola is given by \( y^2 = 4x \). The condition for tangents from a point \( (x_1, y_1) \) to the parabola is given by the equation \( y^2 - 4x = 0 \). For the point \( (k + 3, k) \), we substitute \( x_1 = k + 3 \) and \( y_1 = k \) into the equation.

### Step 2: Substitute the point into the equation
Substituting \( x = k + 3 \) and \( y = k \) into the equation \( y^2 - 4x = 0 \):
\[ k^2 - 4(k + 3) > 0 \]

### Step 3: Simplify the inequality
Now, we simplify the inequality:
\[ k^2 - 4k - 12 > 0 \]

### Step 4: Factor the quadratic expression
Next, we factor the quadratic expression \( k^2 - 4k - 12 \):
\[ k^2 - 6k + 2k - 12 > 0 \]
This can be factored as:
\[ (k - 6)(k + 2) > 0 \]

### Step 5: Determine the critical points
The critical points from the factors are \( k = 6 \) and \( k = -2 \).

### Step 6: Test intervals around the critical points
We will test the intervals defined by these critical points: \( (-\infty, -2) \), \( (-2, 6) \), and \( (6, \infty) \).

1. For \( k < -2 \) (e.g., \( k = -3 \)):
\[ (-3 - 6)(-3 + 2) = (-9)(-1) > 0 \quad \text{(True)} \]

2. For \( -2 < k < 6 \) (e.g., \( k = 0 \)):
\[ (0 - 6)(0 + 2) = (-6)(2) < 0 \quad \text{(False)} \]

3. For \( k > 6 \) (e.g., \( k = 7 \)):
\[ (7 - 6)(7 + 2) = (1)(9) > 0 \quad \text{(True)} \]

### Step 7: Write the solution
From the tests, we find that the inequality \( (k - 6)(k + 2) > 0 \) holds true for:
\[ k \in (-\infty, -2) \cup (6, \infty) \]

### Final Answer
The exhaustive set of values of \( k \) for which the tangents drawn from the point \( (k + 3, k) \) to the parabola \( y^2 = 4x \) are real is:
\[ (-\infty, -2) \cup (6, \infty) \]