Distance between the points `(a cos 35^(@), 0) and B(0,a cos 65^(@))` is

Options

A. `a^(2)`
B. `sqrt(a^(2))`
C. `-a`
D. a

Answer

To find the distance between the points \( A(a \cos 35^\circ, 0) \) and \( B(0, a \cos 65^\circ) \), we can use the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B**:
- Point A has coordinates \( (a \cos 35^\circ, 0) \).
- Point B has coordinates \( (0, a \cos 65^\circ) \).

2. **Substitute the coordinates into the distance formula**:
- Here, \( x_1 = a \cos 35^\circ \), \( y_1 = 0 \), \( x_2 = 0 \), and \( y_2 = a \cos 65^\circ \).
- Therefore, the distance \( d \) becomes:
\[ d = \sqrt{(0 - a \cos 35^\circ)^2 + (a \cos 65^\circ - 0)^2} \]

3. **Simplify the expression**:
- This simplifies to:
\[ d = \sqrt{(-a \cos 35^\circ)^2 + (a \cos 65^\circ)^2} \]
- Which can be rewritten as:
\[ d = \sqrt{(a \cos 35^\circ)^2 + (a \cos 65^\circ)^2} \]

4. **Factor out \( a^2 \)**:
- We can factor out \( a^2 \) from the square root:
\[ d = \sqrt{a^2 (\cos^2 35^\circ + \cos^2 65^\circ)} = a \sqrt{\cos^2 35^\circ + \cos^2 65^\circ} \]

5. **Use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \)**:
- Note that \( \cos 65^\circ = \sin 25^\circ \) and \( \cos 35^\circ = \sin 55^\circ \).
- We can also use the identity \( \cos^2 35^\circ + \cos^2 65^\circ = 1 \) since \( 35^\circ + 65^\circ = 90^\circ \).
- Thus, we have:
\[ \cos^2 35^\circ + \cos^2 65^\circ = 1 \]

6. **Final calculation**:
- Therefore, substituting back, we get:
\[ d = a \sqrt{1} = a \]

### Conclusion:
The distance between the points \( A(a \cos 35^\circ, 0) \) and \( B(0, a \cos 65^\circ) \) is \( a \).