Question

Find the ratio in which the join of A(1, 2, 3)  and B(3, 1, 2) is divided by the plane 2x - y + z = 4
1. 2:3
2. 1:2
3. 1:3
4. 3:2

Answer 1

Correct Answer - Option 3 : 1:3

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 AB be divided by the plane at R in the ratio k:1, 

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

Now, R lies on the plane , so this point must satisfy the equation 2x - y + z = 4

∴ \(\rm \frac{6k+2}{k+1}-\frac{k+2}{k+1}+\frac{2k+3}{k+1}=4\)

\(\rm ⇒ \frac{7k+3}{k+1}=4\)

⇒ 7k + 3 = 4k + 4

⇒ 3k = 1

⇒ k = 1/3

So, the ratio is \(\frac1 3 : 1\)= 1:3

Hence, option (3) is correct.