Question

The point A divides the join of $P(-5,1)$ and $Q(3,5)$ in the ratio $k:1$. The number of values of k for which the area of $\Delta$ABC(where B is $(1,5)$ and C is $(7,-2)$) is $2$, is?

• A. $1$
• B. $2$
• C. $0$
• D. Can't determined

Answer 1

The correct option is B $2$

Use section formula,

point $A=(\frac{(3K-5)}{(1+K)},\frac{(5K+1)}{(K+1)})$

now,$A,B,$ and $C$ points form ∆

use area of ∆ formula ,

$ar\Delta =\frac{1}{2}|\frac{(3K-5)}{(K+1)}(5+2)+1(-2-\frac{(5K+1)}{(K+1)})+7(\frac{(5K+1)}{(K+1)}-5)$

$⟹\pm 4=[\frac{(3K-5)\left(7\right)}{(1+K)}+\frac{(-2K-2-5K-1)}{(1+K)}+\frac{7(5K+1-5K-5)}{(1+K)}]$

$⟹\pm 4(1+K)=21K-35-7K-3-28$

$⟹\pm 4(1+K)=14K-66$

now,

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

$⟹4+4K=14K-66$

$⟹70=10K$

$⟹K=7$

again,

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

$⟹-4-4K=14K-66$

$⟹62=18K$

$⟹K=\frac{31}{9}$

hence ,

$K=7,\frac{31}{9}$