Find the values of `beta` so that the point `(0,beta)` lies on or inside the triangle having the sides `3x+y+2=0`; `2x-3y+5=0` `a n d` `x+4y-14=0.`

Options

A. `[(5)/(3),oo)`
B. `(-oo,(7)/(2)]`
C. `[(5)/(3),(7)/(2)]`
D. none of these

Answer

C

Explanation

Let ABC be the given triangle. The coordinates of the vertices of the triangle ABC are marked in fig. 12 . The point `P(0,beta)` will lie inside or on the triangle ABC, if the following three conditions hold simultaneously.
(i) A and P lie on the same side of BC,
(ii) B and P lie on the same side of AC,
(iii) C and P lie on the same side of AB.
Now, A and P will lie on the same side of BC, if
`(3xx2+3+2)(3xx0+ beta+2) gt 0`
`implies 11(beta+2) ge o implies beta + 2 ge 0 implies beta ge -2" "(i)`
B and P will lie on the same side of AC, if
`(-2xx2-3xx4+5)(2xx0-3beta+5) ge 0`
`implies -11(-2beta+5) ge 0 implies 3beta-5 le 0 implies beta ge (5)/(3)" "...(ii)`

`(-1+1xx4-14)(0+4beta-14) ge 0`
`implies -22(2beta-7) ge 0implies 2beta -7 le 0 implies beta le (7)/(2)" "...(iii)`
From (i), (ii) and (iii), we obtain that
`(5)/(3) le beta le (7)/(2) i.e. beta in [(5)/(3),(7)/(2)]`