The average length of all vertical chords of | vedclass

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(A) The length of a vertical chord at a given $x$ is $L(x) = 2y = 2\frac{b}{a}\sqrt{x^2-a^2}$.The average length $A_L$ is given by $\frac{1}{2a-a} \in

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$b\{2 \sqrt{3}-\ln(2+\sqrt{3})\}$

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(A) The length of a vertical chord at a given $x$ is $L(x) = 2y = 2\frac{b}{a}\sqrt{x^2-a^2}$.The average length $A_L$ is given by $\frac{1}{2a-a} \int_a^{2a} L(x) dx = \frac{1}{a} \int_a^{2a} 2\frac{b}{a}\sqrt{x^2-a^2} dx$.$A_L = \frac{2b}{a^2} \int_a^{2a} \sqrt{x^2-a^2} dx$.Using the integral formula $\int \sqrt{x^2-a^2} dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}|$,we evaluate from $a$ to $2a$:$A_L = \frac{2b}{a^2} [\frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln(x+\sqrt{x^2-a^2})]_a^{2a}$.At $x=2a$: $\frac{2a}{2}\sqrt{4a^2-a^2} - \frac{a^2}{2}\ln(2a+\sqrt{3}a) = a^2\sqrt{3} - \frac{a^2}{2}\ln(a(2+\sqrt{3}))$.At $x=a$: $\frac{a}{2}\sqrt{0} - \frac{a^2}{2}\ln(a) = - \frac{a^2}{2}\ln(a)$.Subtracting these: $A_L = \frac{2b}{a^2} [a^2\sqrt{3} - \frac{a^2}{2}\ln(a(2+\sqrt{3})) + \frac{a^2}{2}\ln(a)] = \frac{2b}{a^2} [a^2\sqrt{3} - \frac{a^2}{2}\ln(2+\sqrt{3})] = b[2\sqrt{3}-\ln(2+\sqrt{3})]$.

The average length of all vertical chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ for $a \leq x \leq 2a$ is:

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