Question

The average length of all vertical chords of hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\),  \(a \leq x\leq 2a\), is

• A. b{\(2\sqrt3\)-ln(\(2+\sqrt3\))}
• B. b{\(3\sqrt2\)+ln(\(3+\sqrt2\))}
• C. b{\(2\sqrt5\)-ln(\(2+\sqrt5\))}
• D. b{\(5\sqrt2\)+ln(\(5+\sqrt2\))}

Answer 1

The Correct Option is A

Given :
Equation of the hyperbola : \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.\)
Given a value of x within the interval [a, 2a], the corresponding y values can be determined using the equation of the hyperbola as follows :
\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.\)
Now, Solving for y, we get :
\(y=±b\sqrt{\frac{x^2}{a^2}-1}\).
The length of a vertical chord at a specific x value is found by calculating the difference between the corresponding y values, resulting in :
Length of the Chord = \(2b\sqrt{\frac{x^2}{a^2}-1}\)
To find the average length of all vertical chords from \(x = a \ to\ x = 2a\), we compute the definite integral of the chord length function over this interval and divide by the interval's length :
Average length of chord = \(\frac{1}{2a-a}\int\limits_a^{2a}\sqrt{\frac{x^2}{a^2}-1}dx\)
By Simplifying this, we get :
Average length of chord = \(\frac{2b}{a}\int\limits_a^{2a}\sqrt{\frac{x^2}{a^2}-1}dx\)
Average length of a vertical chord from \(a\  to\ 2a\) :
\(⇒\frac{2\int\limits_a^{2a}ydx}{\int\limits_a^2dx}=\frac{2\int\limits_a^{2a}\frac{b}{a}\sqrt{x^2-a^2}dx}{(x)^{2a}_a}\)
\(⇒\frac{\frac{2b}{a}\int\limits_a^{2a}\sqrt{x^2-a^2}dx}{a}=\frac{2b}{a^2}\int\limits_0^{2a}\sqrt{x^2-a^2}dx\)
\(=\frac{2b}{a^2}\left(\frac{x}{3}\sqrt{x^2-a^2}-\frac{a^2}{2}\text{ln}|x+\sqrt{x^2-a^2}|\right)^{2a}_a\)
\(=\frac{2b}{a^2}\left[\frac{(2a)\sqrt{4a^2-a^2}}{2}-\frac{a^2\text{ln}|2a+\sqrt{4a^2-a^2}|}{2}-\frac{a\sqrt{a^2-a^2}}{2}+\frac{a^2\text{ln}|a+\sqrt{a^2-a^2}|}{2}\right]\)
\(=\frac{2b}{a^2}\left[\sqrt{3}a^2-\frac{a^2\text{ln}|(2+\sqrt3)a|}{2}+\frac{a^2\text{ln}|a|}{2}\right]\)
\(=2b\left(\sqrt3+\frac{\text{ln}|\frac{a}{(2+\sqrt3)a}|}{2}\right)\)
\(=b(2\sqrt3-\text{ln}|2+\sqrt3|)\)
So, the correct option is (A) : b{\(2\sqrt3\)-ln(\(2+\sqrt3\))}