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The question asks us to find all points on the hyperbola \( x^2 - y^2 = 1 \) that are closest to the point \( (0, 2) \).

To solve this, we can proceed by minimizing the distance between a point \((x, y)\) on the hyperbola and the point \((0, 2)\).

### Solution Steps

1. **Distance Formula**:
   The distance \(D\) between any point \((x, y)\) on the curve and \((0, 2)\) is given by:
   \[
   D = \sqrt{x^2 + (y - 2)^2}
   \]
   To simplify, we can minimize \(D^2\) instead of \(D\) since \(D^2\) and \(D\) will reach their minimums at the same points.

   \[
   D^2 = x^2 + (y - 2)^2 = x^2 + y^2 - 4y + 4
   \]

2. **Substitute the Constraint**:
   From the hyperbola equation \( x^2 - y^2 = 1 \), we get \( x^2 = y^2 + 1 \). Substitute this into \(D^2\):
   
   \[
   D^2 = (y^2 + 1) + y^2 - 4y + 4
   \]
   \[
   D^2 = 2y^2 - 4y + 5
   \]

3. **Minimize \(D^2\) as a Function of \(y\)**:
   Now, we treat \(D^2 = 2y^2 - 4y + 5\) as a function of \(y\) and find its minimum by taking the derivative with respect to \(y\) and setting it to zero.
   
   \[
   \frac{d(D^2)}{dy} = 4y - 4 = 0
   \]
   \[
   y = 1
   \]

4. **Find Corresponding \(x\) Values**:
   Substitute \(y = 1\) back into the hyperbola equation \( x^2 - y^2 = 1 \) to find \(x\):
   
   \[
   x^2 - 1^2 = 1
   \]
   \[
   x^2 = 2
   \]
   \[
   x = \pm \sqrt{2}
   \]

5. **Solution Points**:
   The points on the hyperbola closest to \( (0, 2) \) are:
   \[
   \left( \sqrt{2}, 1 \right) \quad \text{and} \quad \left( -\sqrt{2}, 1 \right)
   \]

### Final Answer
The points on the curve \( x^2 - y^2 = 1 \) closest to \( (0, 2) \) are \( \left( \sqrt{2}, 1 \right) \) and \( \left( -\sqrt{2}, 1 \right) \).

Would you like further details or have any questions?

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Here are some related questions to explore further:

1. How would the solution change if we were asked for points farthest from \( (0, 2) \)?
2. Can this method be applied to other conic sections like circles or ellipses?
3. How can we verify the minimum distance is indeed a minimum and not a maximum?
4. What if the point was not on the y-axis, such as \( (1, 2) \)? How would that affect the approach?
5. What happens to the distance if we move the point \( (0, 2) \) further away, like to \( (0, 5) \)?

**Tip:** Minimizing \(D^2\) instead of \(D\) often simplifies calculations and avoids unnecessary square root operations.

Tentukan semua titik pada kurva x^2 - y^2 = 1 terdekat ke (0, 2).

Explore how to find points on the hyperbola x^2 - y^2 = 1 that are closest to the point (0, 2). This problem involves distance minimization using derivatives, constrained by the hyperbola equation, and is ideal for high school students in Grades 11-12.

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