Question

Find the point of interaction in which the line segment, joining the points P(-1, -3, 4) and Q(4, 2, -1) is divided by the xz-plane.
1. (2, 0, 1)
2. (1, 0, 2)
3. (3, 0, 1)
4. (3, 0, 2)

Answer 1

Correct Answer - Option 1 : (2, 0, 1)

Concept:

Section Formula: Section formula is used to determine the coordinate of a point that divides a line into two parts such that ratio of their length is m : n

1. Let P and Q be the given two points (x1, y1, z1) and (x2, y2, z2) respectively and M(x, y, z) be the point dividing the line segment PQ internally in the ratio m: n

2. Internal Section Formula: When the line segment is divided internally in the ration m: n, we use this formula.\(\rm (x, y, z)=(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n})\)

 

Calculation:

Let, PQ be divided by the xz-plane at a point R in the ratio k:1

Then coordinates of R are (\(\rm (\frac{4k-1}{k+1}, \frac{2k-3}{k+1}, \frac{-k+4}{k+1})\))

Now, R lies on the xz-plane, so y-coordinate will be 0

∴\(\rm \frac{2k-3}{k+1}=0\)

⇒ 2k = 3

⇒ k = 3/2

So, point of interaction = R =  (\(\rm (\frac{4(\frac 3 2)-1}{\frac 3 2+1}, \frac{2(\frac 3 2)-3}{\frac 3 2+1}, \frac{-\frac 3 2+4}{\frac 3 2+1})\))

= (\(\frac{10}{5}, 0, \frac 5 5\))

= (2, 0, 1)

Hence, option (1) is correct.