Q.Find the coordinates of the foot of perpendicular from the point $(-1, 3)$ to the line $3x - 4y - 16 = 0$.

Concept: Foot of Perpendicular from a Point to a Line

We use the formula for the foot $(x_1', y_1')$ of the perpendicular from $(x_1, y_1)$ to $ax + by + c = 0$:

Here $(x_1, y_1) = (-1, 3)$, $a = 3$, $b = -4$, $c = -16$.

First compute $ax_1 + by_1 + c$:

$3(-1) + (-4)(3) - 16 = -3 -12 -16 = -31$

Then $a^2 + b^2 = 9 + 16 = 25$, so the common ratio is:

$-\frac{-31}{25} = \frac{31}{25}$

Now find the foot: …

1. The core idea: what does "foot of perpendicular" mean?

When you drop a perpendicular from a point to a line, the foot is the point on the line that is closest to the given point. Geometrically, the line joining the point to the foot is perpendicular to the given line. So we need to find a point $P$ on $3x - 4y - 16 = 0$ such that the segment from $(-1, 3)$ to $P$ is perpendicular to the line.

Why does this matter? Because the shortest distance from a point to a line is along the perpendicular — and the foot is where that perpendicular meets the line.

2. The formula approach (fastest for exams)

There is a direct formula for the foot of the perpendicular from $(x_1, y_1)$ to the line $ax + by + c = 0$:

Let’s apply it step by step.

3. Step-by-step calculation

Step 1: Identify coefficients and the point.

Line: $3x - 4y - 16 = 0$

So $a = 3$, $b = -4$, $c = -16$.

Point: $(x_1, y_1) = (-1, 3)$.

Step 2: Compute $ax_1 + by_1 + c$.

Step 3: Compute $a^2 + b^2$.

Step 4: Find the correction factor $\frac{ax_1 + by_1 + c}{a^2 + b^2}$.

Step 5: Apply the formula.

For the x-coordinate:

Write $-1$ as $-\frac{25}{25}$:

For the y-coordinate:

Write $3$ as $\frac{75}{25}$:

4. Verification (always a good habit)

Check that $\left( \frac{68}{25}, -\frac{49}{25} \right)$ lies on the line: …