Question

The line passing through the points $(5,1,a)$ and $(3,b,1)$ crosses the $yz-$plane at the point $(0,\frac{17}{2},-\frac{13}{2}),$ then which of the following statement(s) is/are correct ?

• A. $2a-3b=0$
• B. $3a-2b=0$
• C. $a+b-ab+14=0$
• D. $a^2-b^2-ab+4=0$

Answer 1

Answer: (A), (C), (D)

$a^2-b^2-ab+4=0$

Given: $A=(5,1,a),B=(3,b,1)$
Let $yz-$ plane divides the segment $AB$ in $\lambda :1$
$\Rightarrow 0=\frac{3\lambda +5}{\lambda +1}\Rightarrow \lambda =-\frac{5}{3}$
The negative ratio indicates that $yz-$ plane divides the segment $AB$ externally in ratio $5:3$
$\Rightarrow \frac{17}{2}=\frac{5b-3}{5-3}\Rightarrow b=4$
and
$-\frac{13}{2}=\frac{5-3a}{5-3}\Rightarrow a=6$