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Rainbow [258]
2 years ago
15

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + 6xy + 12y2 = 28, (2,

1) (ellipse)
Mathematics
1 answer:
navik [9.2K]2 years ago
5 0

Answer:

The equation of the tangent line is y = -\frac{5}{18}\cdot x +\frac{14}{9}.

Step-by-step explanation:

Firstly, we obtain the equation for the slope of the tangent line by implicit differentiation:

2\cdot x + 6\cdot y + 6\cdot x \cdot y' + 24\cdot y \cdot y' = 0

2\cdot (x + 3\cdot y) + 6\cdot (x + 4\cdot y) \cdot y' = 0

6\cdot (x + 4\cdot y) \cdot y' = -2\cdot (x+3\cdot y)

y' = -\frac{1}{3}\cdot \left(\frac{x + 3\cdot y}{x + 4\cdot y} \right) (1)

If we know that (x,y) = (2, 1), then the slope of the tangent line is:

y' = -\frac{1}{3}\cdot \left(\frac{2+3\cdot 1}{2 + 4\cdot 1} \right)

y' =-\frac{5}{18}

By definition of tangent line, we determine the intercept of the line (b):

y = m\cdot x + b

b = y - m\cdot x (2)

If we know that (x,y) = (2,1) and m = -\frac{5}{18}, then the intercept of the tangent line is:

b = 1 - \left(-\frac{5}{18} \right)\cdot (2)

b = \frac{14}{9}

The equation of the tangent line is y = -\frac{5}{18}\cdot x +\frac{14}{9}.

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