Question

c) Find the angle between the x-axis and the tangent to the hyperbola xy = 9 at (3, 3).

Answer 1

Answer on Question #47105 – Math - Analytic Geometry

Problem.

Find the angle between the x-axis and the tangent to the hyperbola $xy=9$ at (3,3)

Solution:

The tangent to the function $y = f(x)$ at the point $(x_0, f(x_0))$ has equation

$$y = f'(x_0)(x - x_0) + f(x_0).$$

Therefore the tangent to the hyperbola $xy = 9$ at (3,3) has equation

$$y = -\frac{9}{3^2}(x - 3) + 3$$

or

$$y = -x + 6.$$

The tangent of the angle between the $x$-axis and line $y = -x + 6$ equals $-1$. Therefore the angle between this line equal $\arctan(-1) = 135{}^\circ = \frac{3\pi}{4}$.

Answer: $135{}^\circ = \frac{3\pi}{4}$

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