if the points P ( -4,6,10) ,Q ( 2,y,6) and R (14 , 0 ,-2) are such that Q trisects the line segment PR and is nearer to P . find the value of y .

To find the value of \( y \) so that point \( Q \) trisects the line segment between points \( P \) and \( R \), we can start by determining the coordinates of \( Q \) based on the position of \( P \) and \( R \). Since \( Q \) is nearer to \( P \), it will divide the segment \( PR \) into a ratio of 1:2.

Identifying Coordinates of Points

We have the following coordinates:

• P = (-4, 6, 10)
• R = (14, 0, -2)
• Q = (2, y, 6)

Calculating the Trisection Point

Since \( Q \) trisects \( PR \) in the ratio 1:2, we can use the section formula. The coordinates of a point dividing the line segment joining two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in the ratio \( m:n \) are given by:

\[\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right)\]

Assigning Values for Calculation

For points \( P \) and \( R \), we have:

• Coordinates of \( P \): \( (-4, 6, 10) \)
• Coordinates of \( R \): \( (14, 0, -2) \)
• Ratio \( m:n = 1:2 \)

Applying the Section Formula

Let's compute the coordinates of \( Q \) using the section formula.

Calculating x-coordinate

\[x_Q = \frac{1 \cdot 14 + 2 \cdot (-4)}{1 + 2} = \frac{14 - 8}{3} = \frac{6}{3} = 2\]

Calculating y-coordinate

\[y_Q = \frac{1 \cdot 0 + 2 \cdot 6}{1 + 2} = \frac{0 + 12}{3} = \frac{12}{3} = 4\]

Calculating z-coordinate

\[z_Q = \frac{1 \cdot (-2) + 2 \cdot 10}{1 + 2} = \frac{-2 + 20}{3} = \frac{18}{3} = 6\]

Finalizing the Value of y

From our calculations, the coordinates of point \( Q \) are \( (2, 4, 6) \). Therefore, the value of \( y \) is:

y = 4

Summary

To summarize, we used the section formula to determine the coordinates of point \( Q \) that trisects the line segment between points \( P \) and \( R \). Given the ratio of 1:2, we found that \( y \) equals 4, confirming that \( Q \) is indeed positioned correctly as required.