Find the Horizontal Tangent Line x^2+y^2=-18x

Problem

x2+y2=−18*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(−)*18*x)/d(x)

2. Apply the chain rule to the term involving y

2*x+2*yd(y)/d(x)=−18

3. Solve for the derivative d(y)/d(x) to represent the slope.

2*yd(y)/d(x)=−18−2*x

d(y)/d(x)=(−18−2*x)/(2*y)

d(y)/d(x)=(−9−x)/y

4. Set the derivative to zero because horizontal tangent lines have a slope of 0

(−9−x)/y=0

−9−x=0

x=−9

5. Substitute the x-value back into the original equation to find the corresponding yvalues.

(−9)2+y2=−18*(−9)

81+y2=162

y2=81

y=±9

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

y=9

y=−9

Final Answer

y=9,y=−9

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