To determine how many tangent lines to the curve $y = \frac{x}{x + 2}$ pass through the point $(1, 2)$, we need to follow these steps:

Step 1: Derivative of the curve

The slope of the tangent line at any point $x$ on the curve is given by the derivative of $y$: $y = \frac{x}{x + 2}.$ Using the quotient rule: $y' = \frac{(x + 2)(1) - x(1)}{(x + 2)^2} = \frac{2}{(x + 2)^2}.$

Step 2: Equation of the tangent line

The equation of a tangent line to the curve at a point $(x_0, y_0)$ is: $y - y_0 = y'(x_0)(x - x_0).$ Substitute $y_0 = \frac{x_0}{x_0 + 2}$ and $y'(x_0) = \frac{2}{(x_0 + 2)^2}$: $y - \frac{x_0}{x_0 + 2} = \frac{2}{(x_0 + 2)^2}(x - x_0).$

Step 3: Point on the tangent line

The tangent line passes through $(1, 2)$. Substitute $x = 1$ and $y = 2$ into the tangent line equation: $2 - \frac{x_0}{x_0 + 2} = \frac{2}{(x_0 + 2)^2}(1 - x_0).$

Step 4: Simplify the equation

Reorganize the equation: $2 - \frac{x_0}{x_0 + 2} = \frac{2(1 - x_0)}{(x_0 + 2)^2}.$ Multiply through by $(x_0 + 2)^2$ to eliminate the denominator: $2(x_0 + 2)^2 - x_0(x_0 + 2) = 2(1 - x_0).$ Expand the terms: $2(x_0^2 + 4x_0 + 4) - x_0^2 - 2x_0 = 2 - 2x_0.$ $2x_0^2 + 8x_0 + 8 - x_0^2 - 2x_0 = 2 - 2x_0.$ Combine like terms: $x_0^2 + 8x_0 + 8 = 2.$ $x_0^2 + 8x_0 + 6 = 0.$

Step 5: Solve the quadratic equation

Solve $x_0^2 + 8x_0 + 6 = 0$ using the quadratic formula: $x_0 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 1$, $b = 8$, and $c = 6$: $x_0 = \frac{-8 \pm \sqrt{8^2 - 4(1)(6)}}{2(1)} = \frac{-8 \pm \sqrt{64 - 24}}{2}.$ $x_0 = \frac{-8 \pm \sqrt{40}}{2} = \frac{-8 \pm 2\sqrt{10}}{2}.$ $x_0 = -4 \pm \sqrt{10}.$

Step 6: Number of tangent lines

The two solutions $x_0 = -4 + \sqrt{10}$ and $x_0 = -4 - \sqrt{10}$ correspond to two points of tangency on the curve. Thus, there are two tangent lines to the curve that pass through $(1, 2)$.

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Related Questions:

1. How do we interpret the geometric significance of the two tangent lines?
2. What is the equation of the tangent line for a specific solution $x_0$?
3. How can we verify these solutions graphically or numerically?
4. What happens if the point $(1, 2)$ is changed? How does it affect the solution?
5. Can this process be generalized to other rational functions?

Tip:

When solving problems involving tangent lines, always check for symmetry or patterns in the curve; these can simplify computations or confirm solutions.