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

The Correct Option is D

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} $