Find the Tangent Given the Point (6,8)

Problem

x2+y2=100,(6,8)

Solution

1. Identify the equation of the circle. Since the point (6,8) satisfies 6+8=36+64=100 the equation is x2+y2=100

2. Differentiate both sides of the equation with respect to x using implicit differentiation to find the slope of the tangent line.

d(x2)/d(x)+d(y2)/d(x)=d(100)/d(x)

2*x+2*yd(y)/d(x)=0

3. Solve for the derivative d(y)/d(x) which represents the slope m

2*yd(y)/d(x)=−2*x

d(y)/d(x)=−x/y

4. Substitute the coordinates of the given point (6,8) into the derivative to find the specific slope m

m=−6/8

m=−3/4

5. Apply the point-slope formula y−(y_1)=m*(x−(x_1)) using the point (6,8) and the slope m=−3/4

y−8=−3/4*(x−6)

6. Simplify the equation into slope-intercept form.

y−8=−3/4*x+18/4

y=−3/4*x+9/2+8

y=−3/4*x+25/2

Final Answer

y=−3/4*x+25/2

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