The curve C, represented by the given ellipse equation, is obtained by a vertical translation of 3 units from the curve C₁ that passes through the origin. The translation shifts the center of the ellipse upward by 3 units.
To sketch the curve C with the given equation, (x-1)²/16 + (y+1)²/9 = 1, we recognize this as the equation of an ellipse. The center of the ellipse is at the point (1, -1), and the major and minor axes are along the x and y directions, respectively. The semi-major axis is the square root of 16, which is 4, and the semi-minor axis is the square root of 9, which is 3.
The curve C₁, which passes through the origin, is the original ellipse without any translation. For curve C to be obtained by a vertical translation from C₁, we need to determine the value of k. Since the ellipse is symmetric with respect to the x-axis, the vertical translation shifts the entire curve up or down.
The translation of curve C₁ to C involves moving the center (1, -1) vertically by k units. Since k is positive, the translation is upward. Therefore, k is the distance between the original center and the new center. In this case, k is the semi-minor axis of the ellipse.
The exact value of k is the square root of 9, which is 3.