Email
Chat with tutor
logo

Ask Questions, Get Answers

X
 
Questions  >>  CBSE XI  >>  Math  >>  Conic Sections
Answer
Comment
Share
Q)

Find the equation of the hyperbola with foci $(0,\pm\sqrt{10})$,passing through $(2,3)$.

$\begin{array}{1 1}\large\frac{x^2}{5}+\frac{y^2}{5}\normalsize =1\\\large\frac{x^2}{5}-\frac{y^2}{5}\normalsize =1\\\large\frac{x^2}{5}+\frac{y^2}{5}\normalsize =-1\\\large\frac{x^2}{5}-\frac{y^2}{5}\normalsize =-1\end{array} $

1 Answer

Comment
A)
Toolbox:
  • If the vertices of the hyperbola lie on the $y$-axis,then the equation of the hyperbola is $\large\frac{x^2}{a^2}-\frac{y^2}{b^2}$$=-1$ where $a^2=b^2(e^2-1)$.
Answer : $\large\frac{x^2}{5}-\frac{y^2}{5}\normalsize =-1$
Given data is the coordinates of foci are $(0,\pm\sqrt{10})$
The hyperbola passes through the point $(2,3)$.
Hence the equation of the hyperbola is $\large\frac{x^2}{a^2}-\frac{y^2}{b^2}$$=-1$
It passes through $(2,3)$
$\therefore \large\frac{4}{a^2}-\frac{9}{b^2}$$=-1$
But $a^2=b^2(e^2-1)$
$\therefore \large\frac{4}{b^2(e2-1)}-\frac{9}{b^2}$$=-1$------(1)
The coordinates of the foci are $(0,\pm \sqrt{10})$
$\therefore be=\sqrt{10}$
$\Rightarrow b^2e^2=10$
Hence $\large\frac{4}{10-b^2}-\frac{9}{b^2}$$=-1$
On simplifying we get,
$4b^2-9(10-b^2)=-b^2(10-b^2)$
$\Rightarrow 4b^2-90+9b^2=-10b^2+b^4$
$\Rightarrow b^4-23b^2+90=0$
On factorizing we get,
$(b^2-18)(b^2-5)=0$
$\therefore b^2=18$ or $b^2=5$
$a^2=b^2(e^2-1)$
$a^2=10-b^2$
$\;\;\;\;=10-18=-8$ which is not possible.
$a^2=10-5=5$
Hence the required equation is $\large\frac{x^2}{5}-\frac{y^2}{5}$$=-1$
Help Clay6 to be free
Clay6 needs your help to survive. We have roughly 7 lakh students visiting us monthly. We want to keep our services free and improve with prompt help and advanced solutions by adding more teachers and infrastructure.

A small donation from you will help us reach that goal faster. Talk to your parents, teachers and school and spread the word about clay6. You can pay online or send a cheque.

Thanks for your support.
Continue
Please choose your payment mode to continue
Home Ask Homework Questions
Your payment for is successful.
Continue
Clay6 tutors use Telegram* chat app to help students with their questions and doubts.
Do you have the Telegram chat app installed?
Already installed Install now
*Telegram is a chat app like WhatsApp / Facebook Messenger / Skype.
...