Question: A robotics engineer designs a robot whose path follows the hyperbola $ 9x^2 - 36x - 4y^2 + 24y = 36 $. Find the center of the hyperbola.

["Title: Finding the Center of a Hyperbola Design: A Robotics Engineer’s Approach", "When designing robots that navigate complex environments, engineers often rely on precise mathematical models—among them, hyperbolas—to guide path planning and spatial reasoning. In this article, we explore how a robotics engineer determines the center of a hyperbola defined by the equation:", "$$\n9x^2 - 36x - 4y^2 + 24y = 36\n$$", "Understanding the center of the hyperbola is critical, as it represents the geometric midpoint of the curve—essential for aligning robotic sensors, optimizing navigation routes, and predicting future positions.", "### Step 1: Rewriting the Equation in Standard Form", "To identify the center, we first rewrite the given equation by completing the square for both $ x $ and $ y $ terms. The general form of a hyperbola centered at $ (h, k) $ is:", "$$\n\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \quad \ ext{(horizontal transverse axis)}\n$$", "or", "$$\n\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1 \quad \ ext{(vertical transverse axis)}\n$$", "Start with the original equation:", "$$\n9x^2 - 36x - 4y^2 + 24y = 36\n$$", "Group $ x $ and $ y $ terms:", "$$\n(9x^2 - 36x) - (4y^2 - 24y) = 36\n$$", "### Step 2: Completing the Square", "Factor out leading coefficients:", "$$\n9(x^2 - 4x) - 4(y^2 - 6y) = 36\n$$", "Complete the square inside each group:", "- For $ x^2 - 4x $: take half of $-4$ → $-2$, square it → $4$\n- For $ y^2 - 6y $: take half of $-6$ → $-3$, square it → $9$", "Add and subtract inside the parentheses, adjusting the equation:", "$$\n9(x^2 - 4x + 4 - 4) - 4(y^2 - 6y + 9 - 9) = 36\n$$", "$$\n9[(x - 2)^2 - 4] - 4[(y - 3)^2 - 9] = 36\n$$", "Distribute:", "$$\n9(x - 2)^2 - 36 - 4(y - 3)^2 + 36 = 36\n$$", "Simplify:", "$$\n9(x - 2)^2 - 4(y - 3)^2 = 36\n$$", "### Step 3: Dividing to Standard Form", "Divide every term by 36:", "$$\n\frac{9(x - 2)^2}{36} - \frac{4(y - 3)^2}{36} = 1\n\Rightarrow\n\frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1\n$$", "This is now in standard hyperbolic form with center at $ (h, k) = (2, 3) $.", "### Why This Matters for Robotics", "For a robotics engineer, identifying the center is foundational. It allows accurate mapping of the robot’s trajectory relative to the hyperbolic path—critical for collision avoidance, path curvature analysis, and real-time decision-making in robotic navigation systems.", "### Conclusion", "The robot’s designed path follows the hyperbola with center at $ (2, 3) $. By transforming the original equation into standard form, we clearly see the geometric center—demonstrating how mathematical modeling directly supports advanced robotic engineering solutions.", "Keywords: robotics engineer, hyperbola center, robotics navigation, conic sections, mathematical modeling, path planning, center of hyperbola, robotic path design, algebraic geometry in robotics", "Meta Description: Learn how a robotics engineer determines the center of a hyperbola defined by $ 9x^2 - 36x - 4y^2 + 24y = 36 $. Step-by-step solution with modern applications in autonomous robot navigation."]