Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The line is tangent to the hyperbola <img alt="\mathrm{\frac{x^2}{a^2}-\frac

The line \mathrm{2 x+y=1} is tangent to the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is ___________.

Option: 1

2


Option: 2

0


Option: 3

3


Option: 4

1


Answers (1)

best_answer

Point \left(\frac{a}{e}, 0\right) lies on line y=-2 x+1

\mathrm{ \Rightarrow 0=-\frac{2 a}{e}+1 \Rightarrow \frac{a}{e}=\frac{1}{2} \Rightarrow e=2 a \text {. } }

Condition of tangency; c^2=a^2 m^2-b^2

\mathrm{ \begin{aligned} & \Rightarrow \quad 1=4 a^2-b^2 \Rightarrow 1+b^2-4 a^2=0 \, \, \, \, ....(i)\\\\ & \because \quad e^2=1+\frac{b^2}{a^2}=1+\frac{\left(4 a^2-1\right)}{a^2} \end{aligned} }[From (i)]

\mathrm{\begin{aligned} & \Rightarrow e^2=1+4-\frac{1}{a^2} \\\\ & \Rightarrow e^2=5-\frac{4}{e^2} \\\\ & \Rightarrow e^4-5 e^2+4=0 \\\\ & \Rightarrow\left(e^2-1\right)\left(e^2-4\right)=0 \\\\ & \Rightarrow e^2-1 \neq 0, e=2 . \end{aligned}}

Posted by

Deependra Verma

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JAC Chandigarh Counselling 2025: Round 1 seat allotment result out for BTech admissions
anam-khan

JoSAA round 5 seat allotment result 2025 out on josaa.nic.in
anam-khan

BTech aspirant from north-east or UT? Apply for CSAB NEUT counselling 2025 today

Latest Question