Tangent Line E505E5

1. **State the problem:** We are given the implicit function $$y^2 - x^2 = 486$$ and the point $$\left(9, 9\sqrt{7}\right)$$ on its graph. We need to find the slope $$m$$ of the tangent line at this point and then find the equation of that tangent line. 2. **Find the slope using implicit differentiation:** Differentiate both sides of $$y^2 - x^2 = 486$$ with respect to $$x$$: $$\frac{d}{dx}(y^2) - \frac{d}{dx}(x^2) = \frac{d}{dx}(486)$$ Using the chain rule for $$y^2$$ (since $$y$$ is a function of $$x$$): $$2y \frac{dy}{dx} - 2x = 0$$ 3. **Solve for $$\frac{dy}{dx}$$ (the slope $$m$$):** $$2y \frac{dy}{dx} = 2x$$ $$\frac{dy}{dx} = \frac{2x}{2y} = \frac{x}{y}$$ 4. **Evaluate the slope at the point $$\left(9, 9\sqrt{7}\right)$$:** $$m = \frac{9}{9\sqrt{7}} = \frac{1}{\sqrt{7}}$$ 5. **Find the equation of the tangent line:** Use point-slope form: $$y - y_1 = m(x - x_1)$$ Substitute $$m = \frac{1}{\sqrt{7}}$$, $$x_1 = 9$$, and $$y_1 = 9\sqrt{7}$$: $$y - 9\sqrt{7} = \frac{1}{\sqrt{7}}(x - 9)$$ 6. **Final answer:** Slope of tangent line: $$m = \frac{1}{\sqrt{7}}$$ Equation of tangent line: $$y = \frac{1}{\sqrt{7}}(x - 9) + 9\sqrt{7}$$