To find the locus of a point that moves such that its distance from the line \( y = 4 \) is a constant \( k \), we need to understand what it means for a point to maintain a constant distance from a line. ### Step-by-Step Explanation 1. **Understanding the Line**: The line \( y = 4 \) is a horizontal line that runs parallel to the x-axis. Any point on this line has a y-coordinate of 4. 2. **Distance from the Line**: The distance from a point \( (x, y) \) to the line \( y = 4 \) is given by the absolute difference between the y-coordinate of the point and the y-coordinate of the line. Mathematically, this distance can be expressed as: \[ \text{Distance} = |y - 4| \] 3. **Setting the Distance Equal to k**: Since we want the distance from the line to be a constant \( k \), we set up the equation: \[ |y - 4| = k \] 4. **Solving the Absolute Value Equation**: The absolute value equation \( |y - 4| = k \) can be split into two separate equations: - \( y - 4 = k \) - \( y - 4 = -k \) Solving these equations gives us: - From \( y - 4 = k \): \[ y = k + 4 \] - From \( y - 4 = -k \): \[ y = 4 - k \] 5. **Combining the Results**: The two equations \( y = k + 4 \) and \( y = 4 - k \) represent two horizontal lines. Therefore, the locus of the point is the set of all points that lie on these two lines: \[ y = 4 + k \quad \text{and} \quad y = 4 - k \] 6. **Final Locus Representation**: We can express the locus more compactly as: \[ y = 4 \pm k \] This means that the locus consists of two lines: one above \( y = 4 \) and one below \( y = 4 \), separated by a distance of \( k \). ### Evaluating the Options Now, let's evaluate the provided options: - **Option A: \( y = 4 \pm k \)**: This is the correct representation of the locus we derived. It indicates two lines, one at \( y = 4 + k \) and the other at \( y = 4 - k \). - **Option B: \( y = k \pm 4 \)**: This option is incorrect because it suggests lines that are not parallel to the original line \( y = 4 \). Instead, it would represent lines that shift vertically based on the value of \( k \), which does not maintain a constant distance from \( y = 4 \). - **Option C: \( y = 4 + k \)**: This option only represents one of the lines (the one above \( y = 4 \)). It does not account for the line below \( y = 4 \), thus it is incomplete. - **Option D: \( y = k - 4 \)**: This option is incorrect as it does not represent the lines that maintain a constant distance from \( y = 4 \). It shifts the line downwards and does not relate to the original line's distance. ### Summary of Key Points - The locus of points maintaining a constant distance \( k \) from the line \( y = 4 \) consists of two horizontal lines: \( y = 4 + k \) and \( y = 4 - k \). - The correct answer is **Option A: \( y = 4 \pm k \)**. - Understanding absolute values is crucial in determining distances from lines. - Always check each option against the derived equations to ensure they represent the same geometric relationship. This thorough understanding will help you tackle similar problems in the future!