find constant c2:55 pmfor which the line 4y = 2x +c is tangent to the curve y=4x + 8/x
Question
find constant c2:55 pmfor which the line 4y = 2x +c is tangent to the curve y=4x + 8/x
Solution
Sure, let's find the constant c for which the line 4y = 2x + c is tangent to the curve y = 4x + 8/x.
Step 1: Write down the given equations The line equation is 4y = 2x + c, which simplifies to y = 0.5x + c/4. The curve equation is y = 4x + 8/x.
Step 2: Find the derivative of the curve The derivative of y = 4x + 8/x is y' = 4 - 8/x^2.
Step 3: Set the derivative equal to the slope of the line The slope of the line y = 0.5x + c/4 is 0.5. So, we set the derivative equal to 0.5: 0.5 = 4 - 8/x^2.
Step 4: Solve for x Solving the equation 0.5 = 4 - 8/x^2 for x gives x = sqrt(8/7).
Step 5: Substitute x into the curve equation to find y Substituting x = sqrt(8/7) into y = 4x + 8/x gives y = 4sqrt(8/7) + 7sqrt(8/7) = 11sqrt(8/7).
Step 6: Substitute x and y into the line equation to find c Substituting x = sqrt(8/7) and y = 11sqrt(8/7) into y = 0.5x + c/4 gives c = 4y - 2x = 411sqrt(8/7) - 2sqrt(8/7) = 42sqrt(8/7).
So, the constant c for which the line 4y = 2x + c is tangent to the curve y = 4x + 8/x is c = 42sqrt(8/7).
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