Question

In the equation $y-y_1=m(x-x_1)$ if $m$ and $x_1$ are fixed and different lines are drawn from the different value of $y_1$ then

• A. The lines will pass through a single point
• B. There will be a set of parallel lines
• C. There will be one line only
• D. None of these

Answer 1

The correct option is B

There will be a set of parallel lines

Determine the position of lines if $m$ and $x_1$ are fixed in the equation.

As slope $m$ and $x_1$ are fixed then for illustration assume $m=2and{x}_1=2$ then the given line becomes

$\Rightarrow y-y_1=2(x-1)....\left(i\right)$

As line drawn from different values of $y_1$

So taking $y_1=1$ then

$$\begin{gathered} \Rightarrow y-1=2(x-1) \\ \Rightarrow y-2x+1=0....\left(ii\right) \end{gathered}$$

Again taking $y_1=3$ then

$$\begin{gathered} \Rightarrow y-3=2(x-1) \\ \Rightarrow y-2x-1=0....\left(iii\right) \end{gathered}$$

$\because \frac{1}{1}=\frac{-2}{-2}\ne \frac{1}{-1}$ [ condition to check the position of two lines]

Thus, the lines will be parallel, if it is passing from the different value of $y_1$.

Hence, Option B is correct .