If the length of the chord of the circle , ` x^2 + y^2 = r^2 (r gt 0)` along the line , `y - 2x = 3` is `r`, then `r^2` is equal to

Options

A. `9/5`
B. 12
C. `24/5`
D. `12/5`

Answer

To solve the problem, we need to find the value of \( r^2 \) given that the length of the chord of the circle \( x^2 + y^2 = r^2 \) along the line \( y - 2x = 3 \) is equal to \( r \).

### Step-by-Step Solution:

1. **Rewrite the line equation**:
The line equation \( y - 2x = 3 \) can be rewritten as:
\[ y = 2x + 3 \]

2. **Substitute \( y \) into the circle's equation**:
Substitute \( y = 2x + 3 \) into the circle's equation \( x^2 + y^2 = r^2 \):
\[ x^2 + (2x + 3)^2 = r^2 \]

3. **Expand the equation**:
Expanding \( (2x + 3)^2 \):
\[ x^2 + (4x^2 + 12x + 9) = r^2 \]
Combine like terms:
\[ 5x^2 + 12x + 9 - r^2 = 0 \]

4. **Use the quadratic formula**:
The quadratic equation in \( x \) is:
\[ 5x^2 + 12x + (9 - r^2) = 0 \]
The discriminant \( D \) of this quadratic must be non-negative for real intersections:
\[ D = b^2 - 4ac = 12^2 - 4 \cdot 5 \cdot (9 - r^2) \]
Simplifying:
\[ D = 144 - 20(9 - r^2) = 144 - 180 + 20r^2 = 20r^2 - 36 \]

5. **Set the discriminant equal to zero**:
Since the length of the chord is \( r \), we need to find the condition when the length of the chord is equal to \( r \). The length of the chord can be derived from the discriminant:
\[ D = 0 \implies 20r^2 - 36 = 0 \]
Solving for \( r^2 \):
\[ 20r^2 = 36 \implies r^2 = \frac{36}{20} = \frac{9}{5} \]

6. **Final result**:
Therefore, the value of \( r^2 \) is:
\[ \boxed{\frac{9}{5}} \]