A point `A` divides the join of `P(-5,1)` and `Q(3,5)` in the ratio `k :1` . Then the integral value of `k` for which the area of ` A B C ,` where `B` is `(1,5)` and `C` is `(7,-2)` , is equal to 2 units in magnitude is___

Answer

7

Explanation

Using section formula `A-=((3k-5)/(k+1),(5k+1)/(k+1))`.
Area of triangle ABC is 2 sq. units. Therefore,
`(1)/(2)|{:(1,,5,,1),(7,,-2,,1),((3k-5)/(k+1),,(5k+1)/(k+1),,1):}|=+-2`
Operating `R_2toR_2-R_1,R_3toR_3toR_1`, we get
`|{:(1,,5,,1),(6,,-7,,0),((3k-5)/(k+1),,(5k+1)/(k+1)-5,,0):}|=+-4`
or `6((5k+1-5k-5)/(k+1))+7((3k-1-5k-1)/(k+1))=+-4`
or `-24+7(2k-6)=+-4(k+1)`
i.e., `k=7 or k=(31)/(9)`