Find the Horizontal Tangent Line x^2+y^2=-16x

Problem

x2+y2=−16*x

Solution

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

d()/d(x)*(x2+y2)=d()/d(x)*(−16*x)

2*x+2*yd(y)/d(x)=−16

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

2*yd(y)/d(x)=−16−2*x

d(y)/d(x)=(−16−2*x)/(2*y)

d(y)/d(x)=(−8−x)/y

3. Set the derivative to zero because a horizontal tangent line occurs where the slope is zero.

(−8−x)/y=0

−8−x=0

x=−8

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

(−8)2+y2=−16*(−8)

64+y2=128

y2=64

y=±8

5. Identify the equations of the horizontal lines using the calculated yvalues.

y=8

y=−8

Final Answer

y=8,y=−8

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