Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. y^2 − x^2 = 9; (3, 3 root 2 )

Answer 1

2yy' - 2x = 0; y' = x/y = 3/(3√2) = √2/2.

Equation of tangent line: y = 3√2 + √2/2(x - 3); y = √2/2x + 3√2/2

Answer 2

Differentiate the function to determine the slope of the tangent:

y² - x² = 9

2y(^dy/_dx) - 2x = 0

^dy/_dx = ^2x/_2y = ^x/_y

Plug in the point (3, 3√2): ³/_3√2 = ¹/_√2

Plug gradient and point into the general formula for a straight line to calculate b: 3√2 = (¹/_√2)(3) + b

b = ^3√2/₂

Multiplying through by 2 gives you: 2y = √2x + 3√2