15:I[8216,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"BackButton"] 16:I[1553,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"QuestionViewer"] 17:T7a8,The equation of the hyperbola is given by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

For a given value of $x$ in the interval $[a, 2a]$, the corresponding $y$ values can be calculated from the hyperbola equation as follows :

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Solving for $y$, we get $y = \pm b\sqrt{\frac{x^2}{a^2} - 1}$.

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 = $2b\sqrt{\frac{x^2}{a^2} - 1}$

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 = $\frac{1}{2a - a} \int\limits_{a}^{2a} 2b \sqrt{\frac{x^2}{a^2} - 1} dx$

Simplifying this gives us :

Average length of chord = $\frac{2b}{a} \int\limits_{a}^{2a} \sqrt{\frac{x^2}{a^2} - 1} dx$

$\therefore$ Average length of vertical chord from $a$ to $2 a$,

$$ \begin{aligned} & =\frac{2 \int\limits_a^{2 a} y d x}{\int\limits_a^2 d x}=\frac{2 \int\limits_a^{2 a} \frac{b}{a} \sqrt{x^2-a^2} d x}{(x)_a^{2 a}} \\\\ & =\frac{\frac{2 b}{a} \int\limits_a^{2 a} \sqrt{x^2-a^2} d x}{a}=\frac{2 b}{a^2} \int\limits_0^{2 a} \sqrt{x^2-a^2} d x \\\\ & =\frac{2 b}{a^2}\left(\frac{x}{3} \sqrt{x^2-a^2}-\frac{a^2}{2} \ln \left|x+\sqrt{x^2-a^2}\right|\right)_a^{2 a} \\\\ & =\frac{2 b}{a^2}\left[\frac{(2 a) \sqrt{4 a^2-a^2}}{2}-\frac{a^2

Hyperbola

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