The distance of the point (1, 3) from the line 2x + 3y = 6, measured parallel to the line 4x + y = 4, is
Understanding Distance Measured Parallel to a Line
This problem asks for a specific type of distance calculation in coordinate geometry. Instead of the usual perpendicular distance from a point to a line, we need to find the distance measured along a path that is parallel to another given line. Imagine drawing a line through the point (1, 3) that runs parallel to the line 4x + y = 4. This new line will intersect the line 2x + 3y = 6 at some point. The distance we need to find is the length of the line segment between the initial point (1, 3) and this intersection point.
Step-by-Step Solution for Distance Calculation
To find this distance, we will follow these steps:
- Determine the slope of the line parallel to which the distance is measured.
- Find the equation of the line passing through the given point (1, 3) with this slope.
- Calculate the point of intersection of this newly found line and the target line (2x + 3y = 6).
- Use the distance formula to find the distance between the initial point (1, 3) and the intersection point.
Step 1: Finding the Slope of the Parallel Line
The distance is measured parallel to the line 4x + y = 4.
To find the slope of this line, we can rewrite it in the slope-intercept form y = mx + c, where m is the slope.
\[4x + y = 4\] \[y = -4x + 4\]The slope of this line is \(m = -4\). Any line parallel to this line will also have a slope of \(-4\).
Step 2: Equation of the Line Through (1, 3) Parallel to 4x + y = 4
We need the equation of a line that passes through the point \((x_1, y_1) = (1, 3)\) and has a slope \(m = -4\). We can use the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\).
\[y - 3 = -4(x - 1)\]Now, let's simplify this equation:
\[y - 3 = -4x + 4\] \[y = -4x + 4 + 3\] \[y = -4x + 7\]This is the equation of the line passing through (1, 3) and parallel to 4x + y = 4.
Step 3: Finding the Point of Intersection
We need to find where the line \(y = -4x + 7\) intersects the line \(2x + 3y = 6\). We can solve this system of linear equations using substitution.
Substitute the expression for y from the first equation into the second equation:
\[2x + 3(y) = 6\] \[2x + 3(-4x + 7) = 6\]Now, solve for x:
\[2x - 12x + 21 = 6\] \[-10x + 21 = 6\] \[-10x = 6 - 21\] \[-10x = -15\] \[x = \frac{-15}{-10}\] \[x = \frac{3}{2}\]Now that we have the value of x, substitute it back into the equation \(y = -4x + 7\) to find the value of y:
\[y = -4\left(\frac{3}{2}\right) + 7\] \[y = -6 + 7\] \[y = 1\]The point of intersection is \(\left(\frac{3}{2}, 1\right)\).
Step 4: Calculating the Distance Between the Points
Finally, we need to find the distance between the initial point \((x_1, y_1) = (1, 3)\) and the intersection point \((x_2, y_2) = \left(\frac{3}{2}, 1\right)\). We use the distance formula:
\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the coordinates of the two points:
\[\text{Distance} = \sqrt{\left(\frac{3}{2} - 1\right)^2 + (1 - 3)^2}\] \[\text{Distance} = \sqrt{\left(\frac{3}{2} - \frac{2}{2}\right)^2 + (-2)^2}\] \[\text{Distance} = \sqrt{\left(\frac{1}{2}\right)^2 + 4}\] \[\text{Distance} = \sqrt{\frac{1}{4} + 4}\] \[\text{Distance} = \sqrt{\frac{1}{4} + \frac{16}{4}}\] \[\text{Distance} = \sqrt{\frac{1 + 16}{4}}\] \[\text{Distance} = \sqrt{\frac{17}{4}}\]We can simplify the square root:
\[\text{Distance} = \frac{\sqrt{17}}{\sqrt{4}}\] \[\text{Distance} = \frac{\sqrt{17}}{2}\]The distance of the point (1, 3) from the line 2x + 3y = 6, measured parallel to the line 4x + y = 4, is \(\frac{\sqrt{17}}{2}\) units.
The final answer is \(\frac{\sqrt{17}}{2}\) units.
| Point | Coordinates |
|---|---|
| Initial Point | (1, 3) |
| Intersection Point | (\(\frac{3}{2}\), 1) |
Revision Table: Key Concepts Review
Let's quickly review the key concepts used in solving this coordinate geometry problem.
| Concept | Description | Formula/Rule |
|---|---|---|
| Slope of a Line | Measures the steepness of a line. For \(Ax + By = C\), slope is \(-A/B\). For \(y = mx + c\), slope is \(m\). | \(m = \frac{y_2 - y_1}{x_2 - x_1}\) (given two points) |
| Parallel Lines | Lines in the same plane that never intersect. They have the same slope. | \(m_1 = m_2\) for parallel lines |
| Point-Slope Form | Equation of a line given a point \((x_1, y_1)\) and a slope \(m\). | \(y - y_1 = m(x - x_1)\) |
| System of Linear Equations | Two or more linear equations involving the same variables. Solving finds the point(s) where the lines intersect. | Substitution or Elimination method |
| Distance Formula | Calculates the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a Cartesian plane. | \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) |
Additional Information: Different Types of Distance from a Point to a Line
Understanding the different ways distance is measured in coordinate geometry is crucial. This problem highlighted distance measured parallel to another line, but the most common type is perpendicular distance.
- Perpendicular Distance: This is the shortest distance from a point to a line. It's measured along a line segment that is perpendicular to the given line and connects the point to the line. The formula for the perpendicular distance from \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is \(\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}\).
- Distance Measured Parallel to Another Line: As seen in this problem, this distance is measured along a specific direction defined by the parallel line. It involves finding the intersection point of the line through the point (parallel to the given direction) and the target line, then calculating the distance between the two points.
Always read the question carefully to determine which type of distance is being asked for.
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