Find the Horizontal Tangent Line x^2+y^2=-2x

Problem

x2+y2=−2*x

Solution

1. Differentiate implicitly with respect to x to find the slope of the tangent line.

d(x2)/d(x)+d(y2)/d(x)=(d(−)*2*x)/d(x)

2*x+2*yd(y)/d(x)=−2

2. Solve for the derivative d(y)/d(x) by isolating the term.

2*yd(y)/d(x)=−2−2*x

d(y)/d(x)=(−2−2*x)/(2*y)

d(y)/d(x)=(−(1+x))/y

3. Set the derivative to zero because horizontal tangent lines occur where the slope is zero.

(−(1+x))/y=0

−(1+x)=0

x=−1

4. Substitute the x-value back into the original equation to find the corresponding ycoordinates.

(−1)2+y2=−2*(−1)

1+y2=2

y2=1

y=±1

5. Identify the equations of the horizontal lines using the ycoordinates found.

y=1

y=−1

Final Answer

y=1,y=−1

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