We are asked to find the values of $k$ such that the line $y = x + k$ is tangent to the curve $y = 7x - kx^2$.

Step 1: Set up the system

For tangency, the line and the curve must intersect at exactly one point, and at that point, the slopes of the line and the curve must be equal.

1. Equating the two equations (Intersection condition):

We want the line to intersect the curve, so we set their $y$-coordinates equal:

$x + k = 7x - kx^2$

Rearranging this equation:

$0 = -kx^2 + 6x - k$

This gives a quadratic equation in $x$:

$kx^2 - 6x + k = 0$

For tangency, the discriminant of this quadratic equation must be zero (since there is exactly one solution for $x$).

2. Discriminant condition for tangency:

For a quadratic equation $ax^2 + bx + c = 0$, the discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac$

Here, $a = k$, $b = -6$, and $c = k$. Substituting into the discriminant formula:

$\Delta = (-6)^2 - 4(k)(k)$

$\Delta = 36 - 4k^2$

For tangency, $\Delta = 0$:

$36 - 4k^2 = 0$

$4k^2 = 36$

$k^2 = 9$

$k = \pm 3$

Step 2: Conclusion

The values of $k$ such that the line $y = x + k$ is tangent to the curve $y = 7x - kx^2$ are:

$k = 3 \quad \text{or} \quad k = -3$

Would you like more details on any step or have any questions?

Related Questions:

1. What is the general condition for tangency between a line and a curve?
2. How do you find the point of tangency between two functions?
3. How does the discriminant help in determining the nature of roots in a quadratic equation?
4. Can you explain the geometric interpretation of the discriminant?
5. How do we find the slope of a curve at a given point?

Tip:

Always check the discriminant of a quadratic equation when looking for the number of intersection points between a line and a curve.