Question

Find the slope of the tangent to the following curve y²=8(x-6) at the point (8,-4)?

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{curve}\mathrm{is}, \\ {\mathrm{y}}^2=8\left(\mathrm{x}-6\right) \\ \Rightarrow {\mathrm{y}}^2=8\mathrm{x}-48 \\ \mathrm{Differentiating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}\mathrm{x},\mathrm{we}\mathrm{get} \\ 2\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}=8 \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{8}{2\mathrm{y}} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{4}{\mathrm{y}} \\ \mathrm{Now},{\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]}_{\left(8,-4\right)}=\frac{4}{-4}=-1 \\ \mathrm{So},\mathrm{slope}\mathrm{of}\mathrm{the}\mathrm{tangent}\mathrm{to}\mathrm{the}\mathrm{curve}\mathrm{is}-1. \end{gathered}$$