Question

If the foot of the perpendicular drawn from the origin to the line $y=mx+c$ is $(1,1)$ then the value of $m$ and $c$ are, respectively,

• A. 1 and 2
• B. 1 and -2
• C. -1 and 2
• D. 2 and 2
• E. -1 and -2

Hint:

Logic Tip: A point $(1,1)$ satisfying $y = mx + c$ immediately means $m + c = 1$. Checking the given options, only Option B ($1 - 2 = -1$) and Option C ($-1 + 2 = 1$) are mathematically possible. Calculating just the slope narrows it down instantly.

Answer 1

The Correct Option is C

Concept:
If two lines are perpendicular, the product of their slopes is $-1$ ($m_1 \cdot m_2 = -1$). Additionally, if a point is the "foot" of a perpendicular on a line, that point must satisfy the equation of that line.
Step 1: Find the slope of the perpendicular line segment.
The perpendicular is drawn from the origin $O(0,0)$ to the point $P(1,1)$ on the line. The slope of the line segment $OP$ is: $$m_{OP} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{1 - 0} = 1$$
Step 2: Determine the slope of the given line (m).
The line $y = mx + c$ is perpendicular to the segment $OP$. Therefore, its slope $m$ must satisfy: $$m \cdot m_{OP} = -1$$ $$m \cdot 1 = -1 \implies m = -1$$
Step 3: Find the y-intercept (c).
Since the point $P(1,1)$ lies on the line $y = mx + c$, we can substitute $x = 1, y = 1,$ and $m = -1$ into the equation: $$1 = (-1)(1) + c$$ $$1 = -1 + c$$ $$c = 2$$ Thus, $m = -1$ and $c = 2$.