Question

A rectangle is formed by lines \( x = 0, y = 0, x = 3 \) and \( y = 4 \). A line perpendicular to \( 3x + 4y + 6 = 0 \) divides the rectangle into two equal parts. Then the distance of the line from the point \( \left( -1, \frac{3}{2} \right) \) is:

• A. \( \frac{17}{10} \)
• B. \( \frac{10}{17} \)
• C. \( \frac{15}{17} \)
• D. \( \frac{18}{17} \)

Hint:

To find the distance from a point to a line, use the distance formula and substitute the coordinates of the point and the coefficients from the equation of the line.

Answer 1

The Correct Option is A

Step 1: Equation of the Line.
The equation of the given line is: \[ 3x + 4y + 6 = 0 \] We need to find the equation of the line perpendicular to this line.
Step 2: Perpendicular Line's Equation.
The slope of the line \( 3x + 4y + 6 = 0 \) is \( -\frac{3}{4} \), so the slope of the perpendicular line will be \( \frac{4}{3} \) (since the slopes of perpendicular lines are negative reciprocals of each other).
Step 3: Midpoint of the Rectangle.
The rectangle has its vertices at \( (0, 0), (3, 0), (3, 4), (0, 4) \). The midpoint of the rectangle is the average of the coordinates of any two opposite corners. This is: \[ \left( \frac{0 + 3}{2}, \frac{0 + 4}{2} \right) = \left( \frac{3}{2}, 2 \right) \] The perpendicular line divides the rectangle into two equal parts, so it passes through this midpoint.
Step 4: Equation of the Perpendicular Line.
Using the slope \( \frac{4}{3} \) and the midpoint \( \left( \frac{3}{2}, 2 \right) \), the equation of the perpendicular line is: \[ y - 2 = \frac{4}{3} \left( x - \frac{3}{2} \right) \]
Step 5: Find the Distance from the Point.
Now, we need to find the distance from the point \( \left( -1, \frac{3}{2} \right) \) to the line \( 3x + 4y + 6 = 0 \). The formula for the distance from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] Substituting \( A = 3 \), \( B = 4 \), \( C = 6 \), and \( (x_1, y_1) = \left( -1, \frac{3}{2} \right) \), we get: \[ d = \frac{|3(-1) + 4\left( \frac{3}{2} \right) + 6|}{\sqrt{3^2 + 4^2}} = \frac{|-3 + 6 + 6|}{5} = \frac{17}{10} \] Final Answer: \[ \boxed{\frac{17}{10}} \]