Final answer:
To find the equation of the tangent line to the curve at the point (1,1), we use implicit differentiation and the point-slope form of a line. The equation of the tangent line is y - 1 = (-7/13)(x - 1).
Step-by-step explanation:
The student has asked how to use implicit differentiation to find the equation of the tangent line to the given ellipse at the point (1, 1). To do this, we differentiate both sides of the equation of the ellipse with respect to x, treating y as a function of x (y = y(x)). The equation of the ellipse provided is 3x2 + xy + 3y2 = 7.
Differentiating both sides with respect to x:
6x + y + x(dy/dx) + 6y(dy/dx) = 0
Solving for dy/dx (the slope of the tangent line):
dy/dx = -(6x + y) / (x + 6y)
Now, plugging in the point (1, 1) into the above derivative to find the slope at that point:
dy/dx at (1, 1) = -(6(1) + 1) / (1 + 6(1)) = -7/7 = -1
The slope of the tangent line at the point (1, 1) is -1. Using the point-slope form of a line (y - y1 = m(x - x1)), where m is the slope and (x1, y1) is the point (1, 1), we get:
y - 1 = -1(x - 1)
y = -x + 2 (This is the equation of the tangent line)