Find the Tangent Given the Point (3,3)

Problem

x2+y2=18,(3,3)

Solution

1. Identify the equation of the circle and the given point ((x_1),(y_1))=(3,3)

2. Differentiate both sides of the equation x2+y2=18 with respect to x using implicit differentiation.

d(x2)/d(x)+d(y2)/d(x)=d(18)/d(x)

3. Apply the power rule and chain rule to find the derivative.

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

4. Solve for the slope d(y)/d(x) by isolating the term.

2*yd(y)/d(x)=−2*x

d(y)/d(x)=−x/y

5. Evaluate the slope m at the point (3,3)

m=−3/3=−1

6. Substitute the point (3,3) and the slope m=−1 into the point-slope formula y−(y_1)=m*(x−(x_1))

y−3=−1*(x−3)

7. Simplify the equation into slope-intercept form.

y−3=−x+3

y=−x+6

Final Answer

y=−x+6

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