A curve has equation y= kx²+2x-k and a line has equation y=kx-2, where k is a constant. Find the set of values of k for which the curve and line do not intersect.

To find the set of values of \( k \) for which the curve \( y = kx^2 + 2x - k \) and the line \( y = kx - 2 \) do not intersect, we need to determine when these two equations have no common solutions.

### Step 1: Set the equations equal to each other
Since we want to find where they do not intersect, let's set the two equations equal to each other to find where they might intersect:

\[
kx^2 + 2x - k = kx - 2
\]

### Step 2: Rearrange the equation
Rearrange this equation to set it equal to zero:

\[
kx^2 + 2x - k - kx + 2 = 0
\]

Combine like terms:

\[
kx^2 + (2 - k)x + (2 - k) = 0
\]

### Step 3: Analyze the quadratic equation
Now, we have a quadratic equation:

\[
kx^2 + (2 - k)x + (2 - k) = 0
\]

For the curve and line to **not intersect**, this quadratic equation should have **no real solutions**. This happens when the discriminant is less than zero.

### Step 4: Calculate the discriminant
The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\[
\Delta = b^2 - 4ac
\]

In our equation, \(a = k\), \(b = 2 - k\), and \(c = 2 - k\). Substituting these into the discriminant formula, we get:

\[
\Delta = (2 - k)^2 - 4(k)(2 - k)
\]

### Step 5: Set up the inequality
For the quadratic to have no real solutions, the discriminant must be less than zero:

\[
(2 - k)^2 - 4k(2 - k) < 0
\]

### Step 6: Expand and solve the inequality
First, expand both terms:

\[
(2 - k)^2 = 4 - 4k + k^2
\]

\[
4k(2 - k) = 8k - 4k^2
\]

Substituting these into the inequality:

\[
4 - 4k + k^2 - (8k - 4k^2) < 0
\]

Combine like terms:

\[
4 - 4k + k^2 - 8k + 4k^2 < 0
\]

\[
5k^2 - 12k + 4 < 0
\]

### Step 7: Solve the quadratic inequality
To solve \( 5k^2 - 12k + 4 < 0 \), we first find the roots of the equation:

\[
5k^2 - 12k + 4 = 0
\]

Using the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[
k = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 5 \cdot 4}}{2 \cdot 5}
\]

\[
k = \frac{12 \pm \sqrt{144 - 80}}{10}
\]

\[
k = \frac{12 \pm \sqrt{64}}{10}
\]

\[
k = \frac{12 \pm 8}{10}
\]

\[
k = \frac{20}{10} \text{ or } \frac{4}{10}
\]

\[
k = 2 \text{ or } k = 0.4
\]

### Step 8: Determine the intervals
The quadratic \( 5k^2 - 12k + 4 \) is a parabola opening upwards. Thus, it is less than zero between its roots:

\[
0.4 < k < 2
\]

### Conclusion
The set of values of \( k \) for which the curve \( y = kx^2 + 2x - k \) and the line \( y = kx - 2 \) do not intersect is:

\[
k \in (0.4, 2)
\]

### Would you like to see more details or have any other questions? 

### Related Questions:

1. What happens when \( k = 0.4 \) or \( k = 2 \)?
2. How can we find the points of intersection for other values of \( k \)?
3. How would the solution change if the line equation were different?
4. What is the geometric interpretation of the discriminant in this context?
5. How do we analyze the case where the curve and line intersect at exactly one point?

**Tip:** Always double-check your discriminant calculations when working with quadratic equations to ensure accurate results.

Discover how to find the set of values of k for which the quadratic curve y = kx^2 + 2x - k and the line y = kx - 2 do not intersect. This detailed solution covers quadratic equations, discriminant analysis, and inequalities, suitable for Grades 10-12 students.

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