If x=a 2 t(1+ 2 t), y=b 2 t(1- 2 t) then d y d x is equal to

Step-by-step solution

The derivative dy dx is found using parametric differentiation: dy dx = dy/dt dx/dt . Given x = a 2t (1 + 2t) and y = b 2t (1 - 2t) , we compute the derivatives. Differentiating x using the product rule: dx dt = a[(2 2t)(1 + 2t) + 2t(-2 2t)] = 2a[ 2t + ( ^2 2t - ^2 2t)] Applying the double angle identity 4t = ^2 2t - ^2 2t : dx dt = 2a( 2t + 4t) Using the sum-to-product identity: 2t + 4t = 2 3t (-t) = 2 3t t Thus, dx dt = 4a 3t t Differentiating y similarly: dy dt = b[(-2 2t)(1 - 2t) + 2t(2 2t)] = 2b[- 2t + 2 2t 2t