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

Answer: (A)

b{$2\sqrt3$-ln($2+\sqrt3$)}

Answer 2

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($2+\sqrt3$)}