Find the distance of the line $4x - y = 0$ from the point P (4, 1) measured along the line making an angle of $135^\circ$ with the positive x-axis.

Step 1: Find the equation of the line passing through P with a given angle

We are given a point P $(4, 1)$ and an angle of $135^\circ$ that the line makes with the positive $x$ -axis. The slope $m$ of a line is given by the tangent of the angle it makes with the positive $x$ -axis. We can use the point-slope form of a linear equation to find the equation of this line.

Step 2: Substitute values and simplify the equation

Substitute the coordinates of point P $(x_1, y_1) = (4, 1)$ and the slope $m = -1$ into the point-slope form. Then, rearrange the equation to the standard form $Ax + By + C = 0$.

Step 3: Find the intersection point of the two lines

To find the point where the two lines intersect, we need to solve the system of linear equations formed by the given line $4x - y = 0$ and the line we just found, $x + y - 5 = 0$. We can use the elimination method to solve for $x$ and $y$.

Step 4: Calculate the y-coordinate of the intersection point

Now that we have the $x$ -coordinate of the intersection point, we can substitute it back into either of the original line equations to find the corresponding $y$ -coordinate. Using the simpler equation $y = 4x$ from the first line, we find $y = 4$. So, the intersection point is Q $(1, 4)$.

Step 5: Calculate the distance between the two points

Finally, we need to find the distance between the initial point P $(4, 1)$ and the intersection point Q $(1, 4)$. We use the distance formula to calculate this length.

Step 6: Substitute coordinates and compute the distance

Substitute the coordinates of P $(4, 1)$ and Q $(1, 4)$ into the distance formula and simplify the expression to find the final distance.

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