The eccentricity of the curve represented by x=3( t + t ) , y=4( t- t) is

Step-by-step solution

The parametric equations x = 3( t + t) and y = 4( t - t) describe an ellipse in Cartesian form. From x/3 = t + t , squaring yields x^2/9 = 1 + (2t) . Similarly, y/4 = t - t gives y^2/16 = 1 - (2t) upon squaring. Adding these results eliminates the trigonometric term: x^2/9 + y^2/16 = 2 . Dividing by 2 produces the standard ellipse equation x^2/18 + y^2/32 = 1 . With a^2 = 32 and b^2 = 18 , the major axis lies along the y-axis. The eccentricity is e = 1 - b^2/a^2 = 1 - 18/32 = 7/16 = 7 4 . Final answer: 7 4