Let's determine the coordinate of the point P given that it lies on the line segment joining the points (3,2,−1) and (6,2,−2), with the x coordinate of P being 5 . The coordinates of the point P that divides the line segment joining (x1,y1,z1)=(3,2,−1) and (x2,y2,z2)=(6,2,−2) can be found using the section formula. Assume P divides the line segment in the ratio k:1. Therefore, the coordinates of P can be represented as:
P(x,y,z)=(‌
kx2+x1
k+1
,‌
ky2+y1
k+1
,‌
kz2+z1
k+1
) Given that the x coordinate of P is 5 , we write:
‌
kx2+x1
k+1
=5 Plugging x1=3 and x2=6 into the equation, we get: ‌
6k+3
k+1
=5 Solve for k : ‌6k+3=5(k+1) ‌6k+3=5k+5 ‌6k−5k=5−3 ‌k=2
Using this value of k, let's find the y coordinate of P : y=‌
ky2+y1
k+1
Substitute k=2,y1=2, and y2=2 : y=‌
2â‹…2+2
2+1
=‌
4+2
3
=2 Therefore, the y coordinate of P is 2 . The correct answer is: Option A: 2