Find the Horizontal Tangent Line x^2+y^2=-12x

Problem

x2+y2=−12*x

Solution

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

d(x2)/d(x)+d(y2)/d(x)=(d(−)*12*x)/d(x)

2*x+2*yd(y)/d(x)=−12

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

2*yd(y)/d(x)=−12−2*x

d(y)/d(x)=(−12−2*x)/(2*y)

d(y)/d(x)=(−6−x)/y

3. Set the derivative to zero because horizontal tangent lines occur where the slope d(y)/d(x)=0

(−6−x)/y=0

−6−x=0

x=−6

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

(−6)2+y2=−12*(−6)

36+y2=72

y2=36

y=±6

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

y=6

y=−6

Final Answer

Horizontal Tangent Lines: *y=6,y=−6

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