Correct Answer – Option 3 : 5

Concept:

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

Let A and B be the given two points (x1, y1, z₁) and (x2, y2, z₂) respectively and C(x, y, z) be the point dividing the line- segment AB internally in the ratio m: n

I. Internal Section Formula: When the line segment is divided internally in the ratio m : n, we use this formula.

 \((x,\ y,\ z) = \ (\frac{mx_2\ +\ nx_1}{m\ +\ n},\ \frac{my_2\ +\ ny_1}{m\ +\ n},\ \frac{mz_2\ +\ nz_1}{m\ +\ n})\)

II. External Section Formula: When point C lies on the external part of the line segment.

 \((x,\ y,\ z) = \ (\frac{mx_2\ -\ nx_1}{m\ -\ n},\ \frac{my_2\ -\ ny_1}{m\ -\ n},\ \frac{mz_2\ -\ nz_1}{m\ -\ n})\)

Calculation:

Given that,

y-z plane divides the line joining the points (3, 1, 5) and (-2, -1, 4) in the ratio p/q.

We know that when the line segment is divided internally in the ratio m : n, we use the formula.

 \((x,\ y,\ z) = \ (\frac{mx_2\ +\ nx_1}{m\ +\ n},\ \frac{my_2\ +\ ny_1}{m\ +\ n},\ \frac{mz_2\ +\ nz_1}{m\ +\ n})\)

⇒ \(x \ =\ \frac{3p\ +\ (-2q)}{p\ +\ q}\) 

But, in the y-z plane, the x coordinate is zero. 

\(⇒\ x \ =\ \frac{3p\ -\ 2q}{p\ +\ q}\ =\ 0\)

⇒ 3p – 2q = 0

⇒ p/q = 2/3

Therefore, y-z plane devices given line in ratio of 2 : 3.

⇒ p + q = 2 + 3 = 5

Hence, option 3 is correct.