Find the point on the line \(y = 2x + 1\) that is closest to the point \((3, 4)\).
["# How to Find the Point on the Line (y = 2x + 1) Closest to the Point ((3, 4))", "When tasked with finding the point on the line (y = 2x + 1) that is closest to a given point ((3, 4)), understanding geometry and distance formulas is key. This article guides you step-by-step through solving this classic optimization problem using algebra—perfect for students, teachers, and anyone interested in coordinate geometry.", "---", "## Why the Closest Point Matters", "The closest point from a point to a line lies along the perpendicular segment connecting the two. This principle stems from geometry: the shortest distance from a point to a line is always along the perpendicular, not along any other direction.", "---", "## Step 1: Understand the Line and the Point", "We are given:", "- The line:\n [\n y = 2x + 1\n ]\n This is a straight line with slope (m = 2) and y-intercept ((0, 1)).", "- The point:\n ((3, 4))", "Our goal is to find a point ((x, y)) on the line such that the distance between ((x, y)) and ((3, 4)) is minimized.", "---", "## Step 2: Use the Perpendicular Slope", "Since the shortest distance is along the perpendicular to the line:", "- The slope of the given line is (2), so the perpendicular slope is the negative reciprocal, which is\n [\n m_{\perp} = -\frac{1}{2}\n ]", "- The perpendicular line passing through ((3, 4)) has slope (-\frac{1}{2}) and equation:\n [\n y - 4 = -\frac{1}{2}(x - 3)\n ]\n Simplifying:\n [\n y = -\frac{1}{2}x + \frac{3}{2} + 4 = -\frac{1}{2}x + \frac{11}{2}\n ]", "---", "## Step 3: Find the Intersection Point", "The closest point on the line lies at the intersection of:", "1. Original line:\n [\n y = 2x + 1\n ]\n2. Perpendicular line:\n [\n y = -\frac{1}{2}x + \frac{11}{2}\n ]", "Set the two expressions for (y) equal:", "[\n2x + 1 = -\frac{1}{2}x + \frac{11}{2}\n]", "Multiply both sides by 2 to eliminate fractions:", "[\n4x + 2 = -x + 11\n]", "Add (x) and subtract 2 from both sides:", "[\n5x = 9 \quad \Rightarrow \quad x = \frac{9}{5}\n]", "Now substitute (x = \frac{9}{5}) into the original line equation to find (y):", "[\ny = 2\left(\frac{9}{5}\right) + 1 = \frac{18}{5} + 1 = \frac{18}{5} + \frac{5}{5} = \frac{23}{5}\n]", "---", "## Step 4: Final Answer — Closest Point", "The point on the line (y = 2x + 1) closest to ((3, 4)) is:", "[\n\left( \frac{9}{5},\ \frac{23}{5} \right)\n]", "---", "## Bonus: Verify Distance (Optional)", "To confirm, calculate:", "- Distance from ((3, 4)) to (\left(\frac{9}{5}, \frac{23}{5}\right)):\n [\n \sqrt{\left(3 - \frac{9}{5}\right)^2 + \left(4 - \frac{23}{5}\right)^2} = \sqrt{\left(\frac{6}{5}\right)^2 + \left(-\frac{3}{5}\right)^2} = \sqrt{\frac{36 + 9}{25}} = \sqrt{\frac{45}{25}} = \sqrt{\frac{9}{5}}\n ]", "This confirms the calculation is consistent.", "---", "## Why This Method Works", "This approach leverages linear algebra and geometry:", "- Using perpendicular slopes ensures the shortest path by orthogonality.\n- Solving simultaneous equations finds precise intersection.", "No calculus needed—ideal for algebraic problem-solving.", "---", "## SEO Keywords", "Optimize your search:\n``\nclosest point on liney = 2x + 1to point(3, 4)`,\nfind point on line linear closest to (3,4),\ndistance from point to line,\nperpendicular point calculation,\nalgebraic solution closest point line,\nHow to find closest point to a line geometrically", "---", "Summary:**\nTo find the closest point on (y = 2x + 1) to ((3, 4)), draw the perpendicular line through ((3, 4)) using its negative reciprocal slope, solve for their intersection, and you find the closest point (\left( \frac{9}{5}, \frac{23}{5} \right)). This clean, geometric method ensures minimal distance and clears confusion in coordinate geometry."]
