If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is `(x^(2)-4x+y+8)/(x-2)`, then this curve also passes through the point:

Options

A. (5, 4)
B. (4, 5)
C. (4, 4)
D. (5, 5)

Answer

To solve the problem, we need to find the equation of the curve given the slope of the tangent at any point \((x, y)\) and the fact that the curve passes through the origin \((0, 0)\).

### Step-by-Step Solution:

1. **Identify the given slope**:
The slope of the tangent to the curve at any point \((x, y)\) is given by:
\[ \frac{dy}{dx} = \frac{x^2 - 4x + y + 8}{x - 2} \]

2. **Rearranging the equation**:
We can rewrite the equation as:
\[ \frac{dy}{dx} - \frac{y}{x - 2} = \frac{x^2 - 4x + 8}{x - 2} \]
This is a first-order linear differential equation of the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \(P(x) = -\frac{1}{x - 2}\) and \(Q(x) = \frac{x^2 - 4x + 8}{x - 2}\).

3. **Finding the integrating factor**:
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int P(x) \, dx} = e^{\int -\frac{1}{x - 2} \, dx} = e^{-\ln|x - 2|} = \frac{1}{|x - 2|} \]
We can drop the absolute value since \(x \neq 2\) in our context.

4. **Multiplying through by the integrating factor**:
Multiply the entire differential equation by the integrating factor:
\[ \frac{1}{x - 2} \frac{dy}{dx} - \frac{y}{(x - 2)^2} = \frac{x^2 - 4x + 8}{(x - 2)^2} \]

5. **Integrating both sides**:
The left-hand side becomes:
\[ \frac{d}{dx}\left(\frac{y}{x - 2}\right) \]
So we have:
\[ \frac{d}{dx}\left(\frac{y}{x - 2}\right) = \frac{x^2 - 4x + 8}{(x - 2)^2} \]
Integrating both sides gives:
\[ \frac{y}{x - 2} = \int \frac{x^2 - 4x + 8}{(x - 2)^2} \, dx + C \]

6. **Solving the integral**:
We can simplify the integral:
\[ \frac{x^2 - 4x + 8}{(x - 2)^2} = \frac{(x - 2)^2 + 4}{(x - 2)^2} = 1 + \frac{4}{(x - 2)^2} \]
Thus, we can integrate:
\[ \int \left(1 + \frac{4}{(x - 2)^2}\right) \, dx = x - 2 - \frac{4}{x - 2} + C \]

7. **Substituting back**:
Now substituting back, we have:
\[ \frac{y}{x - 2} = x - 2 - \frac{4}{x - 2} + C \]
Multiplying through by \(x - 2\):
\[ y = (x - 2)^2 - 4 + C(x - 2) \]

8. **Using the initial condition**:
Since the curve passes through the origin \((0, 0)\):
\[ 0 = (0 - 2)^2 - 4 + C(0 - 2) \]
Simplifying gives:
\[ 0 = 4 - 4 - 2C \implies 2C = 0 \implies C = 0 \]
Thus, the equation simplifies to:
\[ y = (x - 2)^2 - 4 \]

9. **Finding the points**:
The equation can be rewritten as:
\[ y = x^2 - 4x \]
Now we check the points given in the options to see which one satisfies this equation.

- For \((4, 4)\):
\[ y = 4^2 - 4 \cdot 4 = 16 - 16 = 0 \quad \text{(not satisfied)} \]
- For \((4, 5)\):
\[ y = 4^2 - 4 \cdot 4 = 16 - 16 = 0 \quad \text{(not satisfied)} \]
- For \((5, 5)\):
\[ y = 5^2 - 4 \cdot 5 = 25 - 20 = 5 \quad \text{(satisfied)} \]
- For \((5, 4)\):
\[ y = 5^2 - 4 \cdot 5 = 25 - 20 = 5 \quad \text{(not satisfied)} \]

Thus, the curve passes through the point \((5, 5)\).

### Final Answer:
The curve also passes through the point \((5, 5)\).