Find the distance of the point (3, -5) from the line $3x - 4y - 26 = 0$.

Step 1: Identify the given point and line equation

We are given a point $P(x_1, y_1) = (3, -5)$ and a linear equation $3x - 4y - 26 = 0$. We need to find the perpendicular distance from this point to the given line. The general form of a linear equation is $Ax + By + C = 0$. By comparing, we can identify the coefficients $A$, $B$, and $C$.

Step 2: Recall the distance formula

The formula to calculate the perpendicular distance $d$ from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is given by the absolute value of $Ax_1 + By_1 + C$ divided by the square root of $A^2 + B^2$. This formula is derived from geometric principles and vector projections.

Step 3: Substitute the values into the formula

Now, we substitute the values of $x_1 = 3$, $y_1 = -5$, $A = 3$, $B = -4$, and $C = -26$ into the distance formula. This step sets up the calculation for the numerator and the denominator.

Step 4: Calculate the numerator

We first calculate the expression inside the absolute value in the numerator. This involves performing the multiplications and then the additions and subtractions. The absolute value ensures the distance is always positive.

Step 5: Calculate the denominator

Next, we calculate the value of the denominator. This involves squaring $A$ and $B$, adding the results, and then taking the square root. This part of the formula represents the magnitude of the normal vector to the line.

Step 6: Compute the final distance

Finally, we divide the calculated numerator by the calculated denominator to find the distance $d$. The distance is a scalar value representing the shortest distance from the point to the line.

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