Question

Find a,b,c,d so that the line $x=ay+b,z=cy+d$ must pass through the points $(3,1,-3)(4,2,-4)$ and also show that the given points and $(5,3,-5)$ are collinear.

Answer 1

Let $A(3,1,-3)$, $B(4,2,-4)$ and $C(5,3,-5)$ are points

Line $1$:

$x=ay+b$ ………..$\left(1\right)$

Put $A\&B$ lie on line

$3=a+b$

and $4=2a+b$

$\Rightarrow a=1,b=2$

Line $2$:

$z=cy+d$

Put $A\&B$ lie on line

$-3=c+d$

and $-4=2c+d$

$c=-1$, $d=-2$

Put $C$ in $\left(1\right)$

$5=3\times 1+2=5$

Hence C lies on line $1$ which A & B lie, hence A, B are collinear.