Question

The foot of the perpendicular from the point $(1, 3)$ to the line $x + y - 2 = 0$ is:

• A. $(0, 2)$
• B. $(1, 1)$
• C. $(2, 0)$
• D. $(-1, 3)$

Hint:

You can quickly verify the answer by checking if the coordinate $(0, 2)$ satisfies the line equation: $0 + 2 - 2 = 0$. This eliminates non-satisfying options immediately.

Answer 1

The Correct Option is A

Step 1: Know the foot formula.
The foot of the perpendicular $(h,k)$ from a point $(x_1,y_1)$ to the line $ax + by + c = 0$ follows a neat formula.
\[ \frac{h - x_1}{a} = \frac{k - y_1}{b} = -\frac{ax_1 + by_1 + c}{a^2 + b^2} \]

Step 2: List the numbers.
The point is $(1, 3)$ so $x_1 = 1$, $y_1 = 3$. The line $x + y - 2 = 0$ gives $a = 1$, $b = 1$, $c = -2$.

Step 3: Find the common ratio.
Plug into the right-hand part.
\[ -\frac{(1)(1) + (1)(3) - 2}{1^2 + 1^2} = -\frac{1 + 3 - 2}{2} = -\frac{2}{2} = -1 \]

Step 4: Solve for $h$.
Set $\frac{h - 1}{1} = -1$, so $h - 1 = -1$, giving $h = 0$.

Step 5: Solve for $k$.
Set $\frac{k - 3}{1} = -1$, so $k - 3 = -1$, giving $k = 2$.

Step 6: State the foot.
So the foot of the perpendicular is $(0, 2)$.
\[ \boxed{(0, 2)} \]