Final answer:
To find the equation of the tangent line to the curve 3x² + 8xy - 5y² + 9y = 55 at the point (3,1), we can use implicit differentiation. First, differentiate both sides with respect to x. Then substitute the given point into the equation and solve for y'. Finally, use the point-slope form of the equation of a line to find the equation of the tangent line.
Step-by-step explanation:
To find the equation of the tangent line to the curve 3x² + 8xy - 5y² + 9y = 55 at the point (3,1), we can use implicit differentiation. First, differentiate both sides with respect to x:
6x + 8y + 8xy' - 10yy' + 9y' = 0
Simplifying the equation, we get:
6x + 8y - 10yy' + 8xy' + 9y' = 0
Next, substitute the given point (3,1) into the equation and solve for y':
6(3) + 8(1) - 10(1)y' + 8(3)y' + 9y' = 0
Simplifying further, we find y' = -3/16.
Finally, use the point-slope form of the equation of a line to find the equation of the tangent line, with the slope (y') and the given point (3,1):
y - 1 = (-3/16)(x - 3)
Therefore, the equation of the tangent line to the curve at the point (3,1) is y - 1 = (-3/16)(x - 3).