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.