Consider the curve given by the equation $y^{2}-6 y-9 x^{2}-144 x=576$. It can be shown that $\frac{d y}{d x}=\frac{9(x+8)}{y-3}$.

Find the $x$-coordinate of the point where the line tangent to the curve is the $x$-axis.
$$
x=
$$

Question

Consider the curve given by the equation $y^{2}-6 y-9 x^{2}-144 x=576$. It can be shown that $\frac{d y}{d x}=\frac{9(x+8)}{y-3}$.

Find the $x$-coordinate of the point where the line tangent to the curve is the $x$-axis.
$$
x=
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given the curve $y^{2}-6y-9x^{2}-144x=576$ and its derivative $\frac{dy}{dx}=\frac{9(x+8)}{y-3}$. To find the $x$-coordinate where the tangent line is the $x$-axis, we set $\frac{dy}{dx}=0$. ‖ ‖ step 2 ⋮ Setting the derivative equal to zero gives us $0=\frac{9(x+8)}{y-3}$. This implies that the numerator must be zero since the fraction is equal to zero. ‖ ‖ step 3 ⋮ Solving $9(x+8)=0$ for $x$ gives us $x=-8$. ‖ ∻Answer∻ ⚹ $x=-8$ ⚹ ∻Key Concept∻ ⚹Finding the x-coordinate where the tangent line is the x-axis⚹ ∻Explanation∻ ⚹The x-coordinate is found by setting the derivative equal to zero and solving for $x$.⚹◊What is the derivative of the curve defined by $y^{2}-6y-9x^{2}-144x=576$?⍭ Generate me a similar question◊

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