Two chords of length a unit and b unit of a circle make angles `60^@ ` and `90^@ ` at the centre of a circle respectively, then the correct relation is

Options

A. `b=(3)/(2) a`
B. `b= sqrt(2)a`
C. `b= 2a`
D. `b= sqrt(3) a`

Answer

To solve the problem, we need to establish a relationship between the lengths of the two chords (A and B) and their corresponding angles at the center of the circle (60 degrees and 90 degrees).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the center of the circle be \( O \).
- Let the endpoints of the chord of length \( A \) be \( P \) and \( Q \), making an angle of \( 60^\circ \) at the center \( O \).
- Let the endpoints of the chord of length \( B \) be \( R \) and \( S \), making an angle of \( 90^\circ \) at the center \( O \).

2. **Using the Chord Length Formula**:
- The length of a chord can be calculated using the formula:
\[ \text{Chord Length} = 2R \sin\left(\frac{\theta}{2}\right) \]
- For chord \( A \) (angle \( 60^\circ \)):
\[ A = 2R \sin\left(\frac{60^\circ}{2}\right) = 2R \sin(30^\circ) = 2R \cdot \frac{1}{2} = R \]
- Thus, we have:
\[ R = A \quad \text{(Equation 1)} \]

3. **Calculating for Chord B**:
- For chord \( B \) (angle \( 90^\circ \)):
\[ B = 2R \sin\left(\frac{90^\circ}{2}\right) = 2R \sin(45^\circ) = 2R \cdot \frac{\sqrt{2}}{2} = R\sqrt{2} \]
- Thus, we have:
\[ R = \frac{B}{\sqrt{2}} \quad \text{(Equation 2)} \]

4. **Equating the Two Equations**:
- From Equation 1 and Equation 2, we can equate:
\[ A = \frac{B}{\sqrt{2}} \]
- Rearranging gives us:
\[ B = A\sqrt{2} \]

5. **Conclusion**:
- The relationship between the lengths of the chords \( A \) and \( B \) is:
\[ B = A\sqrt{2} \]

### Final Answer:
The correct relation is \( B = A\sqrt{2} \).