=1. For a given value of x in the interval [a,2a], the corresponding y values can be calculated from the hyperbola equation as follows : ‌
x2
a2
−‌
y2
b2
=1 Solving for y, we get y=±b√‌
x2
a2
−1. The length of a vertical chord at a given x value will be the difference of the y values, which gives us: Length of chord =2b√‌
x2
a2
−1 To calculate the average length of all vertical chords from x=a to x=2a, we take the definite integral of this function over that interval, and divide by the length of the interval : Average length of chord =‌
1
2a−a
‌
2a
∫
a
2b√‌
x2
a2
−1‌dx Simplifying this gives us : Average length of chord =‌