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 slope of the tangent line.

d(x2)/d(x)+d(y2)/d(x)=(d(16)*x)/d(x)

2*x+2*yd(y)/d(x)=16

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

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 horizontal tangent lines occur 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 ycoordinates.

8+y2=16*(8)

64+y2=128

y2=64

y=±8

5. Identify the horizontal lines using the ycoordinates found, ensuring the denominator of the derivative (y is not zero at these points.

y=8

y=−8

Final Answer

y=8,y=−8

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