question_answer

For what value of p, will the following system of linear equations represent parallel lines? \[-x+py=1\]and \[px-y=1\]

A) 2

B) 3

C) 1

D) None of these

Correct Answer: C

Solution :

Given, pair of equations is

\[-x+py-1=0\]                         ...(i)

and    \[px-y-1=0\]                            ...(ii)

On comparing the given equations with standard form i.e. \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] we get

\[{{a}_{1}}=-1,\,{{b}_{1}}=p,\,{{c}_{1}}=-1\]

and      \[{{a}_{2}}=p,\,{{b}_{2}}=-1,\,{{c}_{2}}=-1\]

For parallel lines,

\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\]

(iii)

On taking I and II terms, we get

\[\frac{-1}{\,p}=\frac{p}{-1}\]

\[\Rightarrow \,\,\,{{p}^{2}}=1\,\,\Rightarrow \,p=\pm 1\]

Since, \[p=-1\]does not satisfy the last two terms of Eq. (iii).

\[\therefore \] p = 1 is the required value.

Hence, for p = 1, the given system of equations will represent parallel lines.