Determining the Value of k for a Line Passing Through Origin in the XY-Plane
In the Cartesian coordinate system, the equation of a line can be determined using the slope-intercept form, which is particularly useful when the line passes through the origin. Given two points in the form of (2k, k 32) and the origin (0, 0), we can derive the value of k that satisfies the condition that the line passes through the origin. This article will explore how to determine the value of k using the concept of slope and the properties of the line equation.
Concept of Slope and Line Equation
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
[ m frac{y_2 - y_1}{x_2 - x_1} ]This slope can also be expressed as the ratio of the rise over the run, or the change in y over the change in x.
Determining the Equation of the Line through (2k, k 32)
Given two points (2k, k 32) and the origin (0, 0), we can determine the slope of the line using the slope formula:
[ m frac{(k 32) - 0}{2k - 0} frac{32 - k}{2k} ]Slope Through the Origin
The slope of the line through the origin to any point (0, 0) to (2k, k 32) can also be calculated as follows:
[ m_1 frac{k - 0}{2 - 0} frac{k}{2} ]The same slope through the origin to the point (0, 0) to (k, 32) is:
[ m_2 frac{32 - 0}{k - 0} frac{32}{k} ]Equate the Slopes
Since the line passes through the origin, the slope from the origin to the points (2k, k 32) and (k, 32) must be equal. Therefore, we can set the slopes equal to each other:
[ frac{32 - k}{2k} frac{k}{2} ]Solving for k
Cross-multiplying to eliminate the fractions:
[ 2(32 - k) k(2k) ]Expanding both sides gives:
[ 64 - 2k 2k^2 ]Rearranging the equation:
[ 0 2k^2 - 2k - 64 ]Simplifying further:
[ k^2 - 64 0 ]Factoring:
[ (k - 8)(k 8) 0 ]Solving for k gives:
[ k - 8 0 quad Rightarrow quad k 8 ] [ k 8 0 quad Rightarrow quad k -8 ]Therefore, the possible values of k are 8 and -8.
Example
Let's consider an example. If the points are (20, 8) and the origin (0, 0), the line equation through the points can be determined. The slope from the origin to (20, 8) is:
[ m_1 frac{8 - 0}{20 - 0} frac{8}{20} frac{2}{5} ]The slope from the origin to (0, 32) is:
[ m_2 frac{32 - 0}{k - 0} frac{32}{k} ]Setting the slopes equal:
[ frac{8}{20} frac{32}{k} ]Cross-multiplying:
[ 8k 640 ]Solving for k:
[ k 80 ]Therefore, the line passes through (20, 8) and the origin when k is 80.
Conclusion
In conclusion, by using the concept of slope and the properties of lines, we can determine the value of k for a line passing through the origin in the XY-plane. This method is particularly useful in various fields, including geometry, physics, and engineering, where understanding the relationship between points and lines is crucial.