Question

The point A divides the join of the points (-5,1) and (3,5) in the ratio k:1 and co-ordinates of points B and C are (1,5) and (7,-2) resp. If the area of triangle △ ABC be 2 units, then k equals:
1. 7,9
2. 6,7
3. 7,31/9
4. 9,31/9

Answer 1

P (-5, 1) ==========A =======Q(3,5)

use section formula,

point A ={(3K-5)/(1+K) , (5K+1)/(K+1)}

now, A , B, and C points form ∆
use area of ∆ formula ,

ar∆ =1/2|(3K-5)/(K+1)(5+2) +1{-2-(5K+1)/(K+1)} +7{(5K+1)/(K+1)) -5}


±4 =[ (3K-5)(7)/(1+K) +(-2K-2-5K-1)/(1+K) +7(5K+1-5K-5)/(1+K)]

±4(1+K) ={21K -35 -7K -3 -28 }

±4(1 +K) =14K -66

now,

4( 1 + K) = 14K -66

4 + 4K = 14K -66

70 =10 K

K =7

again,

-4( 1+ K ) = 14K -66

-4 -4K = 14 K -66

62 = 18K

K = 31/9

hence ,
K = 7 , 31/9
so, option (3) is correct

Answer 2

click on thanks for the answer and I hope it will be helpful for u