How to find all points on a curve
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mmm4444bot
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Use the Quadratic Formula to solve the given equation for y. You'll get two expressions in x.
Set the derivative of each equal to -1/2, and solve for x.
Set the derivative of each equal to -1/2, and solve for x.
Hello, pierre!
\(\displaystyle \text{Differentiate implicitly:}\)
. . \(\displaystyle x\tfrac{dy}{dx} + y - 2y\tfrac{dy}{dx} \:=\:0 \quad\Rightarrow\quad 2y\tfrac{dy}{dx} - x\tfrac{dy}{dx} \:=\:y \quad\Rightarrow\quad (2y-x)\tfrac{dy}{dx} \:=\:y\)
. . \(\displaystyle \text{Hence: }\;\frac{dy}{dx} \;=\;\frac{y}{2y-x}\)
\(\displaystyle \text{Then: }\;\frac{y}{2y-x} \;=\;-\frac{1}{2}\)
. . \(\displaystyle \text{Then: }\;2y \;=\;-2y + x \quad\Rightarrow\quad x \:=\:4y\;\;[2]\)
\(\displaystyle \text{Substitute into [1]: }\;(4y)y - y^2 \:=\:1 \quad\Rightarrow\quad 3y^2 \:=\:1 \quad\Rightarrow\quad y^2 \:=\:\frac{1}{3} \quad\Rightarrow\quad y \:=\:\pm\frac{1}{\sqrt{3}}\)
\(\displaystyle \text{Substitute into [2]: }\;x \:=\:4\left(\pm\frac{1}{\sqrt{3}}\right) \:=\:\pm\frac{4}{\sqrt{3}}\)
\(\displaystyle \text{Therefore: }\;\left(\frac{4}{\sqrt{3}},\;\frac{1}{\sqrt{3}}\right) \;\text{ and }\: \left(-\frac{4}{\sqrt{3}},\;-\frac{1}{\sqrt{3}}\right)\)
\(\displaystyle \text{Differentiate implicitly:}\)
. . \(\displaystyle x\tfrac{dy}{dx} + y - 2y\tfrac{dy}{dx} \:=\:0 \quad\Rightarrow\quad 2y\tfrac{dy}{dx} - x\tfrac{dy}{dx} \:=\:y \quad\Rightarrow\quad (2y-x)\tfrac{dy}{dx} \:=\:y\)
. . \(\displaystyle \text{Hence: }\;\frac{dy}{dx} \;=\;\frac{y}{2y-x}\)
\(\displaystyle \text{Then: }\;\frac{y}{2y-x} \;=\;-\frac{1}{2}\)
. . \(\displaystyle \text{Then: }\;2y \;=\;-2y + x \quad\Rightarrow\quad x \:=\:4y\;\;[2]\)
\(\displaystyle \text{Substitute into [1]: }\;(4y)y - y^2 \:=\:1 \quad\Rightarrow\quad 3y^2 \:=\:1 \quad\Rightarrow\quad y^2 \:=\:\frac{1}{3} \quad\Rightarrow\quad y \:=\:\pm\frac{1}{\sqrt{3}}\)
\(\displaystyle \text{Substitute into [2]: }\;x \:=\:4\left(\pm\frac{1}{\sqrt{3}}\right) \:=\:\pm\frac{4}{\sqrt{3}}\)
\(\displaystyle \text{Therefore: }\;\left(\frac{4}{\sqrt{3}},\;\frac{1}{\sqrt{3}}\right) \;\text{ and }\: \left(-\frac{4}{\sqrt{3}},\;-\frac{1}{\sqrt{3}}\right)\)