The ratio in which the line segment joining the points whose position vectors are `2 hat i-4 hat j-7 hat ka n d-3 hat i+5 hat j-8 hat k` is divided by the plane whose equation is ` hat r.( hat i-2 hat j+3 hat k)=13` is a. `13 : 12` internally b. `12 : 25` externally c. `13 : 25` internally d. `37 : 25` internally

Options

A. 13:12 internally
B. 12:25 externally
C. 13:25 internally
D. 37:25 internally

Answer

b

Explanation

Let P be the point and it divides the line segment in the ratio `lamda :1 `. Then,
`" "vec(OP)=vecr= (-3lamda+2)/(lamda+1)hati+ (5lamda-4)/(lamda+1)hatj+ (-8lamda-7)/(lamda+1)hatk`
It satisfies `vecr*(hati-2hatj+3hatk)= 13`. So,
`" "(-3lamda+2)/(lamda+1)-2(5lamda-4)/(lamda+1)+3(-8lamda-7)/(lamda+1)=13`
or `" "-3lamda+2-2(5lamda-4)+3(-8lamda-7)= 13 (lamda+1)`
or `" "-37lamda-11=13lamda+13 or 50 lamda= -24 or lamda= - (12)/(25)`