The equation of the tangent to the curve $x^{2}-y^{2}-8x+2y+11=0$ at (2, 1) is:

Question

Accepted Answer

To determine the equation of the tangent to the curve $x^2 - y^2 - 8x + 2y + 11 = 0$ at the point (2, 1), we need to follow these steps:

1. **Find the derivative of the curve**: The given curve is implicitly defined, so we use implicit differentiation to find $\frac{dy}{dx}$.

Start with the equation:
   $$
   x^2 - y^2 - 8x + 2y + 11 = 0
   $$

Differentiate both sides with respect to $x$:
   $$
   \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2) - \frac{d}{dx}(8x) + \frac{d}{dx}(2y) + \frac{d}{dx}(11) = 0
   $$

This gives:
   $$
   2x - 2y\frac{dy}{dx} - 8 + 2\frac{dy}{dx} = 0
   $$

Rearrange to solve for $\frac{dy}{dx}$:
   $$
   2x - 8 = 2y\frac{dy}{dx} - 2\frac{dy}{dx}
   $$
   $$
   2x - 8 = (2y - 2)\frac{dy}{dx}
   $$
   $$
   \frac{dy}{dx} = \frac{2x - 8}{2y - 2}
   $$

2. **Evaluate the derivative at the point (2, 1)**:
   Substitute $x = 2$ and $y = 1$ into the derivative:
   $$
   \frac{dy}{dx} = \frac{2(2) - 8}{2(1) - 2} = \frac{4 - 8}{2 - 2} = \frac{-4}{0}
   $$

The derivative is undefined, indicating a vertical tangent line at the point (2, 1).

3. **Equation of the tangent line**:
   Since the tangent is vertical, the equation of the tangent line is of the form $x = a$. Given that the tangent passes through the point (2, 1), the equation is $x = 2$.

Thus, the equation of the tangent line is $x - 2 = 0$, which corresponds to Option 4.

**Explanation of why other options are incorrect**:

- **Option 1: $x + 2 = 0$**: This equation represents a vertical line at $x = -2$, which does not pass through the point (2, 1). Therefore, it cannot be the equation of the tangent at (2, 1).

- **Option 2: $2x + 1 = 0$**: This equation simplifies to $x = -\frac{1}{2}$, which is also a vertical line but does not pass through (2, 1). Hence, it is incorrect.

- **Option 3: $x - 2y = 0$**: This equation represents a line with a slope of 1/2, which is not vertical. Since the tangent at (2, 1) is vertical, this option is incorrect.

By following these logical steps, we confirm that Option 4, $x - 2 = 0$, is indeed the correct answer.