To find the value of $k$, we need to determine the coordinates of point $P$ given the conditions and use the section formula.

Given:

• Coordinates of $A$: $(2, 4)$
• Coordinates of $B$: $(10, -12)$
• $P$ lies on the x-axis, meaning the y-coordinate of $P$ is 0.
• $P$ divides $AB$ in the ratio $k : 1$.

Using the section formula, if a point $P(x, y)$ divides a line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m : n$, the coordinates of $P$ are given by: $x = \frac{mx_2 + nx_1}{m+n}$ $y = \frac{my_2 + ny_1}{m+n}$

In our case, the coordinates of $P$ are: $x = \frac{k \cdot 10 + 1 \cdot 2}{k + 1} = \frac{10k + 2}{k + 1}$ $y = \frac{k \cdot (-12) + 1 \cdot 4}{k + 1} = \frac{-12k + 4}{k + 1}$

Since $P$ lies on the x-axis, its y-coordinate is 0: $\frac{-12k + 4}{k + 1} = 0$

Solving for $k$: $-12k + 4 = 0$ $-12k = -4$ $k = \frac{4}{12}$ $k = \frac{1}{3}$

Thus, the value of $k$ is $\frac{1}{3}$.

Would you like more details or have any questions?

Here are 5 questions for you to ask:

1. How do you derive the section formula?
2. Can you show another example of using the section formula?
3. What if the point divides the segment externally?
4. How does the section formula change for 3D coordinates?
5. Can you explain the concept of ratios in coordinate geometry?

Tip: Always verify the conditions given in a problem before applying formulas to ensure accuracy.