Let the ratio in which point $P(4, m)$ divides the line segment joining the points $A(2, 3)$ and $B(6, -3)$ be $k:1$.

Step 1: Use the section formula

The section formula for a point $P(x, y)$ dividing the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $k:1$ is given by: $P\left(\frac{kx_2 + x_1}{k+1}, \frac{ky_2 + y_1}{k+1}\right)$ Substituting the coordinates of $A(2, 3)$, $B(6, -3)$, and $P(4, m)$ into the formula: $P\left(\frac{6k + 2}{k+1}, \frac{-3k + 3}{k+1}\right) = (4, m)$

Step 2: Equate the coordinates

We equate the x-coordinates and y-coordinates separately to find $k$ and $m$.

For the x-coordinate: $\frac{6k + 2}{k+1} = 4$ Cross-multiply to solve for $k$: $6k + 2 = 4(k + 1)$ $6k + 2 = 4k + 4$ $2k = 2 \implies k = 1$

For the y-coordinate: Substitute $k = 1$ into the y-coordinate equation: $\frac{-3(1) + 3}{1+1} = m$ $\frac{-3 + 3}{2} = m \implies m = 0$

Final Answer

The ratio in which $P(4, m)$ divides the line segment joining $A(2, 3)$ and $B(6, -3)$ is $1:1$, and the value of $m$ is $0$.

Would you like more details, or do you have any questions?

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1. What is the section formula and how is it derived?
2. How do you determine the ratio when a point divides a line segment?
3. Can the ratio be negative, and what does that signify geometrically?
4. How do you apply the section formula when the ratio is known?
5. What are some real-world applications of dividing line segments in a given ratio?
6. How does the midpoint formula relate to the section formula?
7. Can this method be used for 3D coordinates?
8. What happens if the coordinates of the points $A$ and $B$ are swapped?

Tip: When working with ratios, always check if the division point lies within the segment or outside, as this will affect the sign of the ratio.