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Let's go through the problem step by step:

### Part (a)
The problem states that the graphs of \( y = x - 1 \) and \( y = kx^2 \) intersect at two distinct points \( A \) and \( B \). We are asked to write down the quadratic equation satisfied by the \( x \)-coordinates of \( A \) and \( B \), and show that \( k < \frac{1}{4} \).

#### Solution for (a):
1. Set the two equations equal to each other to find the intersection points:
   \[
   x - 1 = kx^2
   \]
   Rearranging the terms:
   \[
   kx^2 - x + 1 = 0
   \]
   This is the quadratic equation in \( x \) for the intersection points \( A \) and \( B \).

2. For two distinct intersection points, the discriminant of the quadratic equation must be positive. The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   Here, \( a = k \), \( b = -1 \), and \( c = -1 \). So, the discriminant becomes:
   \[
   \Delta = (-1)^2 - 4(k)(-1) = 1 + 4k
   \]
   For two distinct points, the discriminant must be positive:
   \[
   1 + 4k > 0
   \]
   Solving this inequality:
   \[
   4k < 1 \quad \Rightarrow \quad k < \frac{1}{4}
   \]

So, we've shown that for two distinct points of intersection, \( k \) must be less than \( \frac{1}{4} \).

### Part (b)
We are asked to describe the relationship between the graphs of \( y = x - 1 \) and \( y = kx^2 \) in two cases:
1. \( k = \frac{1}{4} \)
2. \( k > \frac{1}{4} \)

#### Solution for (b):
1. **When \( k = \frac{1}{4} \):**
   - The discriminant \( \Delta = 1 + 4k = 1 + 4\left(\frac{1}{4}\right) = 2 \), which is zero.
   - This means the two graphs touch at exactly one point, i.e., they are tangent to each other at one point.

2. **When \( k > \frac{1}{4} \):**
   - The discriminant \( \Delta = 1 + 4k \) becomes negative.
   - This means there are no real solutions, and hence the graphs do not intersect at all.

### Part (c)
We need to show that for negative values of \( k \), the equation \( x - 1 = kx^2 \) has two real roots, one of which lies between 0 and 1.

#### Solution for (c):
1. Consider the equation again:
   \[
   x - 1 = kx^2 \quad \text{or} \quad kx^2 - x + 1 = 0
   \]
   For \( k < 0 \), the quadratic opens downwards since \( kx^2 \) is negative. This means the parabola will have two real roots (since the discriminant \( 1 + 4k \) is still positive for small negative values of \( k \)).

2. To confirm that one root lies between 0 and 1:
   - When \( x = 0 \), the equation becomes \( -1 = 0 \), which is negative.
   - When \( x = 1 \), the equation becomes \( 1 - 1 = 0 \), which is a root.
   - Therefore, by the Intermediate Value Theorem, since the function is continuous, there must be another root between \( 0 \) and \( 1 \).

Would you like further clarification or more details on any part? Here are some follow-up questions to explore this concept more deeply:

1. How does the shape of the parabola \( y = kx^2 \) change as \( k \) varies?
2. What does the discriminant tell us about the number and nature of the solutions?
3. Can you find the exact points of intersection for specific values of \( k \)?
4. How would the analysis change if \( k \) were negative but less than \( -1 \)?
5. What is the geometrical interpretation of the roots in part (c)?

### Tip:
When analyzing quadratic equations in geometry problems, always check the discriminant to determine the number of intersection points!

The diagram shows the graphs of y = x - 1 and y = kx^2, where k is a positive constant. The graphs intersect at two distinct points A and B. (a) Write down the quadratic equation satisfied by the x-coordinates of A and B, and hence show that k < 1/4. (b) Describe briefly the relationship between the graphs of y = x - 1 and y = kx^2 in each of the cases (i) k = 1/4 (ii) k > 1/4. (c) Show, by means of a graphical argument or otherwise, that when k is a negative constant, the equation x - 1 = kx^2 has two real roots, one of which lies between 0 and 1.

Explore the intersection of the graphs y = x - 1 and y = kx^2 to derive a quadratic equation and analyze the inequality k < 1/4. Understand the relationship between these functions for different values of k and use graphical arguments to show the existence of real roots.

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