On the xy plane where O is the origin, given points, `A(1, 0), B(0, 1) and C(1, 1)`. Let `P, Q, and R` be moving points on the line `OA, OB, OC` respectively such that `overline(OP)=45t overline((OA)),overline(OQ)=60t overline((OB)),overline(OR)=(1-t) overline((OC))` with `t>0.` If the three points `P,Q and R` are collinear then the value of `t` is equal to

Options

A. `(1)/(106)`
B. `(7)/(187)`
C. `(1)/(100)`
D. none of these

Answer

B

Explanation

Again, it is given that the point P,Q and R are collinear.
`implies PQ=lamdaQR`
`implies 15t(4hati-3hati)=lamda[(1-t)(hati+hatj)-60thatj]`
`implies=lamda[(1-t)hati+(1-61t)hatj]`
`implies (45t)/(t-1)=(60t)/(1-61t)`
`implies (3t)/(t-1)=(4t)/(1-61t)`
`implies3(1-61t)=4(t-1)`
`implies 3-183t=4t-4implies 187t=7`
`therefore t=(7)/(187)`.