Find the Horizontal Tangent Line x^2+y^2=20x

Problem

x2+y2=20*x

Solution

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

d()/d(x)*(x2+y2)=d()/d(x)*(20*x)

2*x+2*yd(y)/d(x)=20

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

2*yd(y)/d(x)=20−2*x

d(y)/d(x)=(20−2*x)/(2*y)

d(y)/d(x)=(10−x)/y

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

(10−x)/y=0

10−x=0

x=10

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

10+y2=20*(10)

100+y2=200

y2=100

y=±10

5. Identify the equations of the horizontal lines passing through these points. Since horizontal lines have the form y=k the values found for y define the lines.

Final Answer

y=10,y=−10

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