Consider the curve given by the equation $x y^{2}-x^{3} y=8$. It can be shown that $\frac{d y}{d x}=\frac{3 x^{2} y-y^{2}}{2 x y-x^{3}}$.

Write the equation of the vertical line that is tangent to the curve.

Question

Consider the curve given by the equation $x y^{2}-x^{3} y=8$. It can be shown that $\frac{d y}{d x}=\frac{3 x^{2} y-y^{2}}{2 x y-x^{3}}$.

Write the equation of the vertical line that is tangent to the curve.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the vertical tangent, we need to identify where the denominator of $\frac{dy}{dx}$ is zero, since vertical tangents occur when the slope of the tangent is undefined (which happens when the denominator is zero). ‖ ‖ step 2 ⋮ Set the denominator of $\frac{dy}{dx}$ equal to zero: $2xy - x^3 = 0$. ‖ ‖ step 3 ⋮ Factor out the common term $x$: $x(2y - x^2) = 0$. ‖ ‖ step 4 ⋮ Solve for $y$ when $x eq 0$: $2y - x^2 = 0 \Rightarrow y = \frac{x^2}{2}$. ‖ ‖ step 5 ⋮ Substitute $y = \frac{x^2}{2}$ into the original equation $xy^2 - x^3y = 8$ to find the $x$-values where the vertical tangent occurs. ‖ ‖ step 6 ⋮ After substitution, we get $x\left(\frac{x^2}{2} ight)^2 - x^3\left(\frac{x^2}{2} ight) = 8$. Simplify this to find the $x$-values. ‖ ‖ step 7 ⋮ Simplifying gives $x\frac{x^4}{4} - \frac{x^5}{2} = 8$. This simplifies to $\frac{x^5}{4} - \frac{2x^5}{4} = 8$, which further simplifies to $-\frac{x^5}{4} = 8$. ‖ ‖ step 8 ⋮ Solve for $x$: $x^5 = -32$. ‖ ‖ step 9 ⋮ Take the fifth root of both sides to find $x$: $x = \sqrt[5]{-32}$. ‖ ‖ step 10 ⋮ The fifth root of $-32$ is $-2$, so $x = -2$. ‖ ‖ step 11 ⋮ The equation of the vertical line is $x = -2$. ‖ ∻Answer∻ ⚹ The equation of the vertical line that is tangent to the curve is $x = -2$. ⚹ ∻Key Concept∻ ⚹Vertical Tangents and Slope⚹ ∻Explanation∻ ⚹Vertical tangents occur where the slope of the curve is undefined, which corresponds to the points where the denominator of the derivative $\frac{dy}{dx}$ is zero.⚹◊What is the equation of the vertical line that is tangent to the curve $xy^{2}-x^{3}y=8$?⍭ Generate me a similar question◊

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