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To prove algebraically that the straight line \(x = 2y + 5\) is a tangent to the circle with equation \(x^2 + y^2 = 5\), we need to check if the line intersects the circle at exactly one point.

### Steps:

1. **Substitute the line equation into the circle equation:**
   The line equation is given as \(x = 2y + 5\). We substitute this into the circle's equation \(x^2 + y^2 = 5\).

   \[
   (2y + 5)^2 + y^2 = 5
   \]

2. **Expand and simplify the equation:**

   Expanding \( (2y + 5)^2 \):

   \[
   (2y + 5)^2 = 4y^2 + 20y + 25
   \]

   Substituting back into the circle equation:

   \[
   4y^2 + 20y + 25 + y^2 = 5
   \]

   Combine like terms:

   \[
   5y^2 + 20y + 25 = 5
   \]

3. **Simplify further:**

   Subtract 5 from both sides:

   \[
   5y^2 + 20y + 25 - 5 = 0
   \]

   \[
   5y^2 + 20y + 20 = 0
   \]

   Now divide the entire equation by 5:

   \[
   y^2 + 4y + 4 = 0
   \]

4. **Solve the quadratic equation:**

   Factor the quadratic:

   \[
   (y + 2)^2 = 0
   \]

   This gives \(y = -2\).

5. **Find the corresponding \(x\)-coordinate:**

   Now, substitute \(y = -2\) into the line equation \(x = 2y + 5\):

   \[
   x = 2(-2) + 5 = -4 + 5 = 1
   \]

6. **Conclusion:**

   The point of intersection is \((1, -2)\). Since the quadratic equation gave a repeated root, the line touches the circle at exactly one point, meaning it is a tangent to the circle.

Thus, the line \(x = 2y + 5\) is indeed a tangent to the circle \(x^2 + y^2 = 5\).

Would you like further details or clarifications?

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Here are 5 related questions:
1. How do you derive the general formula for the distance between a point and a line?
2. What is the geometric interpretation of the discriminant in a quadratic equation?
3. How can you find the equation of a tangent line to a circle at a given point?
4. How do you determine the number of intersections between a line and a circle algebraically?
5. What is the difference between a secant and a tangent to a circle?

**Tip**: Always check the discriminant (\(\Delta\)) when determining if a quadratic equation has one solution (tangent), two solutions (secant), or no solution (no intersection).

Prove algebraically that the straight line with equation x = 2y + 5 is a tangent to the circle with equation x^2 + y^2 = 5.

Discover how to prove algebraically that the line x = 2y + 5 is tangent to the circle with equation x^2 + y^2 = 5. This solution includes detailed steps on substituting the line equation into the circle equation, solving the resulting quadratic equation, and confirming the condition for tangency. Suitable for students in Grades 10-12.

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