For the line \(\mathrm{y}-\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right), \mathrm{m}\) and \(\mathrm{x}_{1}\) are fixed values, if different lines are drawn according to the different value of \(\mathrm{y}_{1}\), then all such lines would be \(\ldots\) (a) all lines intersect the line \(\mathrm{x}=\mathrm{x}_{1}\) (b) all lines pass through one fixed point (c) all lines are parallel to the line \(\mathrm{y}=\mathrm{x}_{1}\) (d) all lines will be the set of perpendicular lines

In order to gain insight into the relationship between these lines, we should first rewrite the given equation in slope-intercept form, i.e., \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To do this, we'll solve for \(y\): \(y = mx - mx_1 + y_1\). Notice now that both \(m\) and \(x_1\) are fixed values, and \(y_1\) can vary. This means that the slope of all these lines will be equal to \(m\), but the y-intercept will vary depending on the value of \(y_1\). #Step 2: Analyze the choices given#

Now, let's examine each choice and see which one is correct based on the equation \(y = mx - mx_1 + y_1\): (a) all lines intersect the line \(x = x_1\): To find this out, we can plug \(x = x_1\) into our equation and see what happens: \(y = m(x_1) - m(x_1) + y_1 \Rightarrow y = y_1\). So, every line intersects the line \(x = x_1\) at the point \((x_1, y_1)\). Thus, this choice is correct. (b) all lines pass through one fixed point: Since the y-intercept varies depending on the value of \(y_1\), the lines do not pass through the same point, and this choice is incorrect. (c) all lines are parallel to the line \(y = x_1\): The line \(y = x_1\) has a slope of 0 and is a horizontal line. The given lines have a fixed slope of \(m\), but no constraint is given on whether \(m\) is 0 or any other value. Therefore, this choice is incorrect, as it cannot be concluded that they are always parallel to the horizontal line \(y = x_1\). (d) all lines will be the set of perpendicular lines: Since all lines have the same fixed slope, \(m\), they cannot be perpendicular to each other, making this choice incorrect. #Conclusion# Based on our analysis, the correct answer is (a): all lines intersect the line \(x = x_1\).