Find the Tangent Line at the Point 2x^2+y^2=12 , (2,-2)

Problem

2*x2+y2=12,(2,−2)

Solution

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

(d(2)*x2)/d(x)+d(y2)/d(x)=d(12)/d(x)

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

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

2*yd(y)/d(x)=−4*x

d(y)/d(x)=−(4*x)/(2*y)

d(y)/d(x)=−(2*x)/y

3. Evaluate the slope m by substituting the given point (2,−2) into the derivative.

m=−(2*(2))/(−2)

m=(−4)/(−2)

m=2

4. Apply the point-slope formula y−(y_1)=m*(x−(x_1)) using the point (2,−2) and the slope m=2

y−(−2)=2*(x−2)

y+2=2*x−4

5. Simplify to slope-intercept form by isolating y

y=2*x−4−2

y=2*x−6

Final Answer

y=2*x−6

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