This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Write down the given equation of the family of ellipses. The equation is: (x^2)/(a^2) + (y^2)/(4) = 1 Here, a is an arbitrary constant. Step 2: Differentiate the equation with respect to x. Since there is one arbitrary constant (a), we differentiate the equation once with respect to x. (d)/(dx)((x^2)/(a^2)) + (d)/(dx)((y^2)/(4)) = (d)/(dx)(1) (1)/(a^2)(2x) + (1)/(4)(2y)(dy)/(dx) = 0 (2x)/(a^2) + (y)/(2)(dy)/(dx) = 0 Step 3: Solve for (1)/(a^2) from the differentiated equation. (2x)/(a^2) = -(y)/(2)(dy)/(dx) (1)/(a^2) = -(y)/(2x) · (1)/(2)(dy)/(dx) (1)/(a^2) = -(y)/(4x)(dy)/(dx) Step 4: Substitute the expression for (1)/(a^2) back into the original equation. Substitute (1)/(a^2) into (x^2)/(a^2) + (y^2)/(4) = 1: x^2 (-(y)/(4x)(dy)/(dx)) + (y^2)/(4) = 1 -(xy)/(4)(dy)/(dx) + (y^2)/(4) = 1 Step 5: Rearrange the equation to match the given options. Multiply the entire equation by 4 to eliminate the denominators: -xy(dy)/(dx) + y^2 = 4 Rearrange the terms to isolate xy(dy)/(dx): xy(dy)/(dx) = y^2 - 4 Comparing this result with the given options: 1. xy(dy)/(dx) = 4 - y^2 2. xy(dy)/(dx) = 4 - x^2 3. xy(dy)/(dx) = x^2 - 4 4. xy(dy)/(dx) = y^2 - 4 The derived differential equation matches option 4. The final answer is 4. xy(dy)/(dx) = y^2 - 4.