Download App

Conic Sections — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsConic Sections3 Marks
Question
If e and e’ are the eccentricities of a hyperbola and its conjugate hyperbola respectively,prove that $\frac{1}{e^2}+\frac{1}{\left(e^{\prime}\right)^2}=1$.
✓

Answer

Let e be the eccentricity of a hyperbola
$\frac{x^2}{ a ^2}-\frac{y^2}{ b ^2}=1$
$e =\frac{\sqrt{ a ^2+ b ^2}}{ a }$
$e ^2=\frac{ a ^2+ b ^2}{ a ^20}$
$\frac{1}{ e ^2}=\frac{ a ^2}{ a ^2+ b ^2}$
$\ldots$..(i)
Also, $e ^{\prime}$ is the eccentricity of conjugate
hyperbola $\frac{y^2}{ b ^2}-\frac{x^2}{ a ^2}=1$
$e ^{\prime}=\frac{\sqrt{ b ^2+ a ^2}}{ b }$
$\left( e ^{\prime}\right)^2=\frac{ b ^2+ a ^2}{ b ^2}$
$\frac{1}{\left(e^{\prime}\right)^2}=\frac{b^2}{b^2+a^2}$
...(ii)
Adding (i) and (ii), we get
$\frac{1}{ e ^2}+\frac{1}{\left( e ^{\prime}\right)^2}=\frac{ a ^2}{ a ^2+ b ^2}+\frac{ b ^2}{ a ^2+ b ^2}=\frac{ a ^2+ b ^2}{ a ^2+ b ^2}$
$\frac{1}{ e ^2}+\frac{1}{\left( e ^{\prime}\right)^2}=1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Conic Sections→Conic Sections — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Test the continuity of the following functions at the points or intervals indicated against them :
$f(x)=\frac{x^2+8 x-20}{2 x^2-9 x+10} \quad$ for $0$
$=12, \text { for } x=2$
$=\frac{2-2 x-x^2}{x-4} \quad \text { for } 3 \leq x<4$
at $x=2$
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{1-\tan\text{x}}{\text{x}-\frac{\pi}{4}}$
→
Find the area of the triangle formed by the lines:
$y = m_1x + c_1, y = m_2x + c_2$ and $x = 0$
→
Find the magnitude, in radians and degrees, of the interior angle of a regular.
Pentagon
→
Check if the following functions have an inverse function. If yes, find the inverse function.

$f(x)=\frac{6 x-7}{3}$

→
How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.
→
Without expanding the determinants, show that

$\left|\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right|=0$

→
Find variance and S.D. for the following set of numbers. 65, 77, 81, 98, 100, 80, 129
→
Express the following matrices as the sum of a symmetric and a skew symmetric matrix.

$\left[\begin{array}{ccc}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2\end{array}\right]$

→
Prove that:
n!(n + 2)! = n! + (n + 1)!
→