Find the Tangent Line at the Point 2x^2+y^2=57 , (-4,5)

Problem

2*x2+y2=57,(−4,5)

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(57)/d(x)

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

2. Isolate the derivative d(y)/d(x) to solve for the slope function.

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 derivative at the given point (−4,5) to find the specific slope m

m=−(2*(−4))/5

m=8/5

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

y−5=8/5*(x−(−4))

y−5=8/5*(x+4)

5. Simplify the equation into slope-intercept form y=m*x+b

y−5=8/5*x+32/5

y=8/5*x+32/5+5

y=8/5*x+32/5+25/5

y=8/5*x+57/5

Final Answer

y=8/5*x+57/5

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