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: Concept
The foot of the perpendicular $(h, k)$ from any point $(x_1, y_1)$ to the line $ax + by + c = 0$ can be calculated using the projection formula: \[ \frac{h - x_1}{a} = \frac{k - y_1}{b} = -\frac{a x_1 + b y_1 + c}{a^2 + b^2} \]

Step 2: Meaning
Here, the point is $(x_1, y_1) = (1, 3)$ and the line coefficients are $a = 1$, $b = 1$, $c = -2$.

Step 3: Analysis
Evaluate the constant ratio: \[ \text{Ratio} = -\frac{1(1) + 1(3) - 2}{1^2 + 1^2} = -\frac{1 + 3 - 2}{2} = -\frac{2}{2} = -1 \] Now solve for $h$ and $k$: \[ \frac{h - 1}{1} = -1 \implies h - 1 = -1 \implies h = 0 \] \[ \frac{k - 3}{1} = -1 \implies k - 3 = -1 \implies k = 2 \] Thus, the foot of the perpendicular is $(0, 2)$.

Step 4: Conclusion
The coordinates of the foot of the perpendicular are $(0, 2)$. Final Answer: (A)