Convert Polar Equation to Rectangular Equation
Converting Polar Equation to Rectangular Equation
20 r = 1 + cos e
Simply the rectangular equation by moving all of the terms to the left side of the equation and combining them with one another.
Final answer: The polar equation r = 16/(1 + cos(θ)) is converted to a rectangular equation by substituting r and θ with their rectangular equivalents. The final result is 2x² - 16x + y² - 256 = 0.
Explanation: To convert the polar equation r = 16/(1 + cos(θ)) to a rectangular equation, we use the relationship between polar and rectangular coordinates, which are (r × cos(θ), r × sin(θ)) = (x, y). Replace r in the polar equation with sqrt(x² + y²) and cos(θ) with x/r, we get: sqrt(x² + y²) = 16/(1 + x/sqrt(x² + y²)). By cross-multiplying and squaring both sides to remove the square root, we can simplify to a rectangular equation: x² + y² = 256 + 16x - x². Moving all terms to the left gives: 2x² - 16x + y² - 256 = 0.
What is the process of converting a polar equation to a rectangular equation? The process of converting a polar equation to a rectangular equation involves substituting the polar variables (r and θ) with their respective rectangular counterparts (x and y), utilizing the relationships between polar and rectangular coordinates, and simplifying the resulting equation to express it in rectangular form.