To find the coordinates of the point on the x-axis, we can use the section formula. The section formula allows us to find the coordinates of a point dividing a line segment in a given ratio.

Problem Breakdown:

• The point $B(-3, 4)$ is given.
• We need to find the point $P(x, 0)$ on the x-axis, where the distance between $P$ and $B$ is in the ratio 1:2.
• The point $P(x, 0)$ divides the segment $OB$ (where $O$ is the origin $(0, 0)$) in the ratio 1:2.

Step 1: Use the section formula

The formula to find the coordinates of a point dividing a line segment in the ratio $m:n$ is:

$P\left( \frac{n x_1 + m x_2}{m+n}, \frac{n y_1 + m y_2}{m+n} \right)$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the points that the segment joins, and $m:n$ is the given ratio.

Step 2: Apply to this case

• The coordinates of $O$ are $(0, 0)$.
• The coordinates of $B$ are $(-3, 4)$.
• The ratio $m:n = 1:2$.

Using the section formula, the coordinates of the point $P(x, 0)$ dividing $OB$ in the ratio 1:2 are:

$x = \frac{2(0) + 1(-3)}{1 + 2} = \frac{-3}{3} = -1$

$y = \frac{2(0) + 1(4)}{1 + 2} = \frac{4}{3} \approx 1.33$

Since we're looking for a point on the x-axis, the correct answer should be $(-1, 0)$.

So the coordinates of the point are (-1, 0). None of the options match this exactly. There could be an error in the choices listed or in interpreting the conditions