Question: Find the point on the line \(y = 2x + 1\) in the plane that is closest to the point \((4, 3)\).
["Finding the Point on the Line (y = 2x + 1) Closest to ((4, 3)): A Step-by-Step Solution", "When tasked with finding the point on a line that is nearest to a given point in the plane, the goal is to identify the perpendicular projection of the point onto the line. In this article, we’ll explore how to determine the closest point on the line (y = 2x + 1) to the point ((4, 3)) using geometry and algebra.", "### Why Use the Perpendicular Line Approach?", "The shortest distance from a point to a line occurs along the line segment that is perpendicular to the original line. This means the closest point lies where the line through ((4, 3)) with slope perpendicular to the original line intersects (y = 2x + 1).", "### Step 1: Identify the slope of the given line\nThe equation of the line is:\n[\ny = 2x + 1\n]\nThis is in slope-intercept form (y = mx + b), where (m = 2).\nSo, the slope of the given line is (2).", "### Step 2: Find the slope of the perpendicular line\nThe slope of a line perpendicular to another is the negative reciprocal.\nIf the original slope is (2), then the perpendicular slope is:\n[\nm_{\perp} = -\frac{1}{2}\n]", "### Step 3: Write the equation of the perpendicular line passing through ((4, 3))\nUsing the point-slope form:\n[\ny - y_1 = m(x - x_1)\n]\nSubstitute ((x_1, y_1) = (4, 3)) and (m = -\frac{1}{2}):\n[\ny - 3 = -\frac{1}{2}(x - 4)\n]\nSimplify:\n[\ny - 3 = -\frac{1}{2}x + 2\n]\n[\ny = -\frac{1}{2}x + 5\n]", "### Step 4: Find the intersection of the two lines\nSet the original line equal to the perpendicular line:\n[\n2x + 1 = -\frac{1}{2}x + 5\n]\nMultiply both sides by 2 to eliminate fractions:\n[\n4x + 2 = -x + 10\n]\nAdd (x) to both sides:\n[\n5x + 2 = 10\n]\nSubtract 2:\n[\n5x = 8 \quad \Rightarrow \quad x = \frac{8}{5} = 1.6\n]\nNow substitute (x = \frac{8}{5}) into the original line equation to find (y):\n[\ny = 2\left(\frac{8}{5}\right) + 1 = \frac{16}{5} + 1 = \frac{21}{5} = 4.2\n]", "### Step 5: The closest point on the line\nThus, the point on the line (y = 2x + 1) closest to ((4, 3)) is:\n[\n\left(\frac{8}{5},\ \frac{21}{5}\right)\n]\nor in decimal form approximately ((1.6,\ 4.2)).", "### Summary\n- The shortest distance point is found by projecting the point perpendicularly onto the line.\n- Use the slope’s negative reciprocal to find the perpendicular line.\n- Solve the system to find the intersection — the closest point.", "---", "Why This Analges helps:\nUnderstanding projection geometry helps in applications like computer graphics, physics, navigation, and optimization. Knowing how to calculate the nearest point on a line is fundamental in both theoretical mathematics and practical problem solving.", "If you're studying geometry, linear algebra, or working on related problems, mastering this technique saves time and deepens conceptual understanding.", "---", "Frequently Asked Questions\nQ: Can I find the closest point using distance formula alone?\nA: While using the distance formula directly leads to a minimization problem, it’s inefficient without calculus. The perpendicular projection method is both faster and more reliable.", "Q: Does this method apply to any line?\nA: Yes, as long as the line is not vertical (which avoids undefined slopes), the perpendicular line provides a valid solution.", "Q: What if the point lies on the line?\nA:** In that case, the closest point is the point itself, so distance is zero — a special case of the projection.", "---", "By following these clear steps and understanding the geometry, you can confidently find the closest point on any line to a given point in the plane.\nIf you found this helpful, share this SEO-optimized guide to master perpendicular projections and distance-minimizing geometry!"]
