$$Question: A particle physicist models the trajectory of a neutrino through a detector as the line $ y = \frac{3}{4}x + 2 $. Find the point on this trajectory that is closest to the detector's origin $ (0, 0) $.

["Finding the Closest Point on a Neutrino Trajectory to the Detector Origin\nAn Analysis Using Geometry and Optimization", "When a particle physicist models the path of a neutrino through a detector as a straight line given by the equation $ y = \frac{3}{4}x + 2 $, a fundamental question arises: What is the point on this line closest to the origin $ (0, 0) $? Solving this problem not only reveals key geometric insights but also supports precision measurements in experimental particle physics.", "---", "### Why Finding the Closest Point Matters", "In detector design, identifying the nearest point on a particle’s trajectory to a sensor station or origin is critical for locating interaction events accurately. This point minimizes the travel distance the neutrino undertakes before detection—minimizing uncertainty and improving signal resolution.", "---", "### The Mathematical Setup", "We seek the point $ (x, y) $ on the line $ y = \frac{3}{4}x + 2 $ that minimizes the Euclidean distance to the origin $ (0, 0) $. The distance $ d $ from any point $ (x, y) $ to $ (0, 0) $ is:", "$$\nd = \sqrt{x^2 + y^2}\n$$", "Since the square root is increasing, minimizing $ d $ is equivalent to minimizing $ d^2 = x^2 + y^2 $. Substituting $ y = \frac{3}{4}x + 2 $:", "$$\nd^2 = x^2 + \left( \frac{3}{4}x + 2 \right)^2\n= x^2 + \left( \frac{9}{16}x^2 + 3x + 4 \right)\n= \left(1 + \frac{9}{16}\right)x^2 + 3x + 4\n= \frac{25}{16}x^2 + 3x + 4\n$$", "---", "### Minimizing the Quadratic Expression", "To find the minimum of $ f(x) = \frac{25}{16}x^2 + 3x + 4 $, take the derivative and set it to zero:", "$$\nf'(x) = \frac{50}{16}x + 3 = \frac{25}{8}x + 3\n$$", "Set $ f'(x) = 0 $:", "$$\n\frac{25}{8}x + 3 = 0 \Rightarrow x = -\frac{3 \cdot 8}{25} = -\frac{24}{25}\n$$", "Now substitute back to find $ y $:", "$$\ny = \frac{3}{4} \left( -\frac{24}{25} \right) + 2 = -\frac{72}{100} + 2 = -\frac{18}{25} + 2 = \frac{32}{25}\n$$", "---", "### Final Answer: The Closest Point", "Thus, the point on the neutrino’s path $ y = \frac{3}{4}x + 2 $ closest to the origin is:", "$$\n\left( -\frac{24}{25},\ \frac{32}{25} \right)\n$$", "This point lies where the line perpendicular to $ y = \frac{3}{4}x + 2 $—with slope $ -\frac{4}{3} $ (negative reciprocal)—intersects the trajectory, confirming geometric rigor in designing precision neutrino detectors.", "---", "### Summary", "- The shortest distance from a point to a line occurs along the perpendicular.\n- Substituting into the trajectory equation solves for the minimizing $ x $.\n- Evaluating $ y $ confirms the closest point.\n- This method applies broadly in physics, engineering, and data analysis involving optimal distance measurement.", "---", "Keywords: neutrino trajectory, closest point on line, distance minimization, particle physics, detector modeling, geometry in physics, negative reciprocal slope, Euclidean distance, mathematical physics.\nMeta Description:\nFind the point on the neutrino path $ y = \frac{3}{4}x + 2 $ closest to the origin $ (0, 0) $ using calculus and geometry. Solution: $ \left( -\frac{24}{25},\ \frac{32}{25} \right) $. Perfect for physics and detector design applications."]