The coordinates of the points A and B are (-4, 9) and (12, -3) respectively. The straight line passing through A and B cuts the x-axis at the point P. (a)
Find AP : BP.
(b)
Find the coordinates of P.

To solve the problem, we'll address each part step by step.

### Part (a) Find \( \frac{AP}{BP} \)

The ratio \( \frac{AP}{BP} \) can be found using the section formula. If the line \( AB \) cuts the x-axis at point \( P(x_1, 0) \), then the coordinates of \( P \) divide the line segment \( AB \) in a certain ratio \( m:n \). The formula for finding the coordinates of the point \( P \) dividing the line joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is:

\[
x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}
\]

Given that \( y = 0 \) at point \( P \) (since \( P \) lies on the x-axis), we use the y-coordinate formula to find the ratio \( m:n = AP:BP \):

\[
0 = \frac{m(-3) + n(9)}{m+n}
\]

\[
0 = -3m + 9n
\]

\[
3m = 9n
\]

\[
\frac{m}{n} = \frac{9}{3} = 3
\]

Thus, \( AP:BP = 3:1 \).

### Part (b) Find the coordinates of \( P \)

Now that we know the ratio \( AP:BP = 3:1 \), we can find the x-coordinate of \( P \) using the section formula for the x-coordinate:

\[
x = \frac{mx_2 + nx_1}{m+n}
\]

Substituting \( m = 3 \), \( n = 1 \), \( x_1 = -4 \), and \( x_2 = 12 \):

\[
x = \frac{3(12) + 1(-4)}{3+1} = \frac{36 - 4}{4} = \frac{32}{4} = 8
\]

So, the coordinates of \( P \) are \( (8, 0) \).

### Summary of the Answers:
- (a) The ratio \( \frac{AP}{BP} = 3:1 \).
- (b) The coordinates of point \( P \) are \( (8, 0) \).

Would you like any further details or explanations?

Here are 5 related questions for further practice:
1. How would you find the equation of the line passing through points \( A \) and \( B \)?
2. What is the slope of the line through \( A \) and \( B \)?
3. If the y-axis cuts the line \( AB \), how do you find the point of intersection?
4. Can you find the length of segment \( AB \)?
5. How do you determine if the point \( (x_3, y_3) \) lies on the line \( AB \)?

**Tip:** The section formula is very useful for dividing a line segment into a specific ratio and finding the coordinates of the dividing point.

Learn how to find the ratio AP to BP and the coordinates of point P on a line passing through points A(-4, 9) and B(12, -3). Detailed step-by-step solution using the section formula and ratio calculations. Suitable for students in Grades 9-12 studying coordinate geometry.

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