Question

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)

Answer 1

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 $y = -\frac{5}{18}\cdot x +\frac{14}{9}$.