Question

Suppose $P(2,\,y,\,z)$ lies on the line through $A(3,-1,4)$ and $B(-4,2,1)$ . Then, the value of z is equal to

• A. $\frac{-1}{2}$
• B. $\frac{19}{4}$
• C. $\frac{-19}{4}$
• D. $\frac{25}{7}$

Answer 1

Answer: (D)

$\frac{25}{7}$

Answer 2

The equation of line passing through the point
$A(3,-1,4)$ and $B(-4,2,1)$ .
$\frac{x-{{x}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{y-{{y}_{1}}}{{{y}_{2}}-{{y}_{1}}}=\frac{z-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}}$
$\Rightarrow$ $\frac{x-3}{-4-3}=\frac{y+1}{2+1}=\frac{z-4}{1-4}$
$\Rightarrow$ $\frac{x-3}{-7}=\frac{y+1}{3}=\frac{z-4}{-3}$
Since, the point $P(2,y,z)$ passing through the above line,
then $\Rightarrow$ $\frac{2-3}{-7}=\frac{y+1}{3}=\frac{z-4}{-3}$
$\Rightarrow$ $\frac{y+1}{3}=\frac{z-4}{-3}=\frac{1}{7}$
$\Rightarrow$ $y=\frac{3}{7}-1$
and $z=-\frac{3}{7}+4$
$\Rightarrow$ $y=-\frac{4}{7}$
and $z=\frac{25}{7}$