Question

The nearest point on the line $x+2y=5$ from the point $P(7,9)$ is equal to

• A. (6,1)
• B. (7,6)
• C. (2,3)
• D. (8,3)
• E. (3,1)

Hint:

Math Tip: A direct formula exists for the foot of the perpendicular $(h, k)$ from a point $(x_1, y_1)$ to a line $ax + by + c = 0$: $$ \frac{h - x_1}{a} = \frac{k - y_1}{b} = -\frac{ax_1 + by_1 + c}{a^2 + b^2} $$ Using this bypasses finding the slope and second equation entirely!

Answer 1

Answer: (E)

Concept:
Coordinate Geometry - Foot of a Perpendicular.
The "nearest point" on a line from a given point is the foot of the perpendicular dropped from that point onto the line.
Step 1: Find the slope of the given line.
Let the given line be $L_1: x + 2y = 5$.
Convert it to slope-intercept form ($y = mx + c$): $$ 2y = -x + 5 $$ $$ y = -\frac{1}{2}x + \frac{5}{2} $$ The slope of $L_1$ is $m_1 = -\frac{1}{2}$.
Step 2: Determine the slope of the perpendicular line.
Let the perpendicular line from $P(7,9)$ be $L_2$.
Since $L_1$ and $L_2$ are perpendicular, the product of their slopes is $-1$ ($m_1 \cdot m_2 = -1$). $$ \left(-\frac{1}{2}\right) \cdot m_2 = -1 $$ $$ m_2 = 2 $$
Step 3: Find the equation of the perpendicular line ($L_2$).
Using the point-slope form $y - y_1 = m(x - x_1)$ with point $P(7,9)$ and slope $m_2 = 2$: $$ y - 9 = 2(x - 7) $$ $$ y - 9 = 2x - 14 $$ Rearrange into standard form: $$ 2x - y = 5 $$
Step 4: Find the intersection point (the nearest point).
Solve the system of linear equations for $L_1$ and $L_2$: 1) $x + 2y = 5$ 2) $2x - y = 5 \implies y = 2x - 5$ Substitute equation (2) into equation (1): $$ x + 2(2x - 5) = 5 $$ $$ x + 4x - 10 = 5 $$ $$ 5x = 15 $$ $$ x = 3 $$
Step 5: Solve for the y-coordinate.
Substitute $x = 3$ back into the expression for $y$: $$ y = 2(3) - 5 $$ $$ y = 6 - 5 = 1 $$ The nearest point is $(3, 1)$.