Question

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

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

Hint:

Foot of perpendicular from $(x_1,y_1)$ to $ax+by+c=0$ is $(x_1-a\frac{ax_1+by_1+c}{a^2+b^2}, y_1-b\frac{ax_1+by_1+c}{a^2+b^2})$.

Answer 1

The Correct Option is B

1. We need to find the foot of the perpendicular from the point \((-2, 3)\) to the line given by the equation \(2x - y - 3 = 0\).
2. First, recall the formula for the perpendicular distance from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\):

\[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

1. The slope of the given line \(2x - y - 3 = 0\) can be obtained by rewriting the equation as \(y = 2x - 3\), so the slope is \(2\).
2. The slope of the perpendicular line is the negative reciprocal of the slope of the given line, which is \(\frac{-1}{2}\).
3. The equation of the line perpendicular to \(2x - y - 3 = 0\) and passing through \((-2, 3)\) is calculated using the point-slope form:

\[ y - 3 = \frac{-1}{2}(x + 2) \]

1. Simplifying, we get:

\[ 2y - 6 = -x - 2 \implies x + 2y - 4 = 0 \]

1. Now, solve the system of equations:
   • \(2x - y - 3 = 0\)
   • \(x + 2y - 4 = 0\)
2. Multiply the second equation by 2 for elimination:

\[ 2x + 4y - 8 = 0 \]

1. Subtract the first equation from the modified second equation:

\[ (2x + 4y - 8) - (2x - y - 3) = 0 \implies 5y - 5 = 0 \implies y = 1 \]

1. Substitute \(y = 1\) into \(x + 2y - 4 = 0\):

\[ x + 2 \times 1 - 4 = 0 \implies x = 2 \]

1. Thus, the foot of the perpendicular is \((2, 1)\).
2. Therefore, the correct answer is \((2, 1)\), which matches the provided correct answer.