Question

A rectangle is formed by the lines \( x = 0,\ y = 0,\ x = 3,\ y = 4 \). 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. \(2\)
• B. \( \frac{17}{10} \)
• C. \( \frac{6}{5} \)
• D. \( \frac{8}{5} \)

Hint:

Any line that divides a rectangle into two equal areas must pass through its centroid.

Answer 1

The Correct Option is B

To solve this problem, we first find the equation of a line that is perpendicular to the given line and divides the rectangle into two equal areas. Then, we calculate the distance of a given point from this line.

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1. Find the slope of the required line
   The given line is: \[ 3x + 4y + 6 = 0 \] Rewriting in slope–intercept form: \[ y = -\frac{3}{4}x - \frac{3}{2} \] Hence, its slope is: \[ m_1 = -\frac{3}{4} \] The slope of a line perpendicular to it is: \[ m_2 = \frac{4}{3} \]
2. Locate the midpoint of the rectangle
   The rectangle is bounded by: \[ x = 0,\; y = 0,\; x = 3,\; y = 4 \] Its midpoint is: \[ \left( \frac{3}{2},\, 2 \right) \] A line passing through this point will divide the rectangle into two equal areas.
3. Equation of the required line
   Using the point–slope form: \[ y - 2 = \frac{4}{3}\left(x - \frac{3}{2}\right) \] Simplifying: \[ 4x - 3y + 1 = 0 \]
4. Distance of the point from the line
   The given point is: \[ \left(-1, \frac{3}{2}\right) \] The distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
5. Substitute values
   \[ d = \frac{|4(-1) - 3\left(\frac{3}{2}\right) + 1|}{\sqrt{4^2 + (-3)^2}} \] \[ = \frac{| -4 - \frac{9}{2} + 1 |}{\sqrt{25}} = \frac{\left|-\frac{17}{2}\right|}{5} \]
6. Simplify
   \[ d = \frac{17}{10} \]

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Final Answer:
The distance of the line from the point \(\left(-1, \frac{3}{2}\right)\) is \[ \boxed{\frac{17}{10}} \]