The distance between the points (a cos 35°, 0) and (0, a cos 65°) can be found using the distance formula in coordinate geometry. Let's break down the problem and solve it step by step.

Given:
Point 1: (a cos 35°, 0)
Point 2: (0, a cos 65°)

Distance Formula:
The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Step 1: Identify the coordinates of the two points
Point 1: (a cos 35°, 0)
Point 2: (0, a cos 65°)

Step 2: Apply the distance formula
Distance = √((0 - a cos 35°)^2 + (a cos 65° - 0)^2)

Step 3: Simplify the expression
Distance = √(a^2 cos^2 35° + a^2 cos^2 65°)

Step 4: Use trigonometric identities to simplify further
Recall the trigonometric identity: cos^2 θ + sin^2 θ = 1
We can rewrite the equation as:
Distance = √(a^2 (cos^2 35° + cos^2 65°))

Step 5: Apply the trigonometric identities
Using the identity cos^2 θ = 1 - sin^2 θ, we can rewrite the equation as:
Distance = √(a^2 (1 - sin^2 35° + 1 - sin^2 65°))
Distance = √(a^2 (2 - sin^2 35° - sin^2 65°))

Step 6: Use the trigonometric identity sin^2 θ = 1 - cos^2 θ
Applying the identity, we have:
Distance = √(a^2 (2 - (1 - cos^2 35°) - (1 - cos^2 65°)))
Distance = √(a^2 (2 - 1 + cos^2 35° + cos^2 65°))
Distance = √(a^2 (1 + cos^2 35° + cos^2 65°))

Step 7: Calculate the values of cos^2 35° and cos^2 65°
Using a scientific calculator, we find that cos^2 35° ≈ 0.814 and cos^2 65° ≈ 0.184

Step 8: Substitute the values into the equation
Distance = √(a^2 (1 + 0.814 + 0.184))
Distance = √(a^2 (1.998))

Step 9: Simplify the expression
Distance = √(1.998a^2)
Distance ≈ 1.414a

Conclusion:
The distance between the points (