Discussion Forum

If points P(4,5,x) , Q(3,y,4) and R (5,8,0) are collinear , then the value of x+y is 

Answer:

Explanation:

Given P(4,5,x), Q (3,y,4) and R(5,8,0) are collinear

 $\therefore$     $\overrightarrow{PQ}=\lambda \overrightarrow{QR}$

$\Rightarrow$   $\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}$

 = $(3\hat{i}+y\hat{j}+4\hat{k})-(4\hat{i}+5\hat{j}+x\hat{k})=-\hat{i}+(y-5)\hat{j}+(4-x)\hat{k}$

 and 

$\overrightarrow{QR}=\overrightarrow{OR}-\overrightarrow{OQ}=(5\hat{i}+8\hat{j})-(3\hat{i}+y\hat{j}+4\hat{k})= 2\hat{i}+(8-y)\hat{j}-4\hat{k}$

 $\therefore$    $-\hat{i}+(y-5)\hat{j}+(4-x)\hat{k}$

 = $\lambda[2\hat{i}+(8-y)\hat{j}-4\hat{k}]$

 On equating  the component of vector both sides, we get,

$\frac{-1}{2}=\lambda,\frac{y-5}{8-y}=\lambda,\frac{4-x}{-4}=\lambda$

 On putting the value of $\lambda$ , we get

$(y-5)=(8-y)(-\frac{1}{2})$

 and   $(4-x)=-4(-\frac{1}{2})$

 $\Rightarrow$   2y-10=y-8

    and 4-x=2

$\Rightarrow$   y=2

 and x=2

 $\therefore$   x+y=2+2=4