Question

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.y² - x²/64 = 1; (6, -5/4)

Accepted Answer

Recognize that the given curve is an implicit function defined by the equation $$y^{2}$$ - \frac{$$x^{2}$$}{64} = 1$$. Since y$ is not explicitly solved for x$, we will use implicit differentiation to find $$\frac{dy}{dx}$$, the slope of the tangent line. Differentiate both sides of the equation with respect to x$. Remember to apply the chain rule when differentiating y$$^{2}$$, treating y$ as a function of x$:

$$\frac{d}{dx}\$$left(y^{2}$$\right) - \frac{d}{dx}\left(\frac{$$x^{2}$$}{64}\right) = \frac{d}{dx}(1)$$

which becomes

2y$$ \cdot \frac{dy}{dx} - \frac{2x}{64} = 0$$. Solve the resulting equation for $$\frac{dy}{dx}$$ to express the slope of the tangent line in terms of x$ and y$:

2y$$ \cdot \frac{dy}{dx} = \frac{2x}{64}$$

so

$$\frac{dy}{dx} = \frac{2x}{64 \cdot 2y} = \frac{x}{64y}$$. Substitute the given point (6$$, -\frac{5}{4})$$ into the expression for $$\frac{dy}{dx}$$ to find the slope of the tangent line at that point:

m$$ = \frac{6}{64 \cdot (-\frac{5}{4})}$$. Use the point-slope form of the equation of a line to write the equation of the tangent line:

y$$ - $$y_1 = m(x$$ - $$x_1$$)$$,

where $$($$x_1$$, $$y_1) = (6$$, -\frac{5}{4})$$ and m$ is the slope found in the previous step.

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.
y² - x²/64 = 1; (6, -5/4)

Verified video answer for a similar problem:

Key Concepts

Implicit Differentiation

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Hyperbola and Its Properties