Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Let are circles of unit radius with centre at <img alt="\math

Let \mathrm{C_1\: and\: C_2} are circles of unit radius with centre at \mathrm{(0,0) \: and \: (1,0)} respectively. \mathrm{C_3} is a circle of unit radius, passes through the centre of the circles \mathrm{C_1\: and \: C_2} and have its centre above x-axis. Equation of the common tangent to \mathrm{C_1\: and \: C_3} which does not pass through \mathrm{C_2} is
 

Option: 1

\mathrm{x-\sqrt{3} y+2=0}
 


Option: 2

\mathrm{\sqrt{3} x-y+2=0}
 


Option: 3

\mathrm{\sqrt{3} x-y-2=0}
 


Option: 4

\mathrm{x+\sqrt{3} y+2=0}


Answers (1)

best_answer

Equation of any circle through \mathrm{(0,0) \: and\: (1,0)} is
\mathrm{ (x-0)(x-1)+(y-0)(y-0)+\lambda\left|\begin{array}{lll} x & y & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \end{array}\right|=0 }

\mathrm{ \Rightarrow \quad x^2+y^2-x+\lambda y=0}

If it represents \mathrm{C_3}, its radius =1

\Rightarrow \quad 1=\left(\frac{1}{4}\right)+\left(\frac{\lambda^2}{4}\right) \Rightarrow \lambda= \pm \sqrt{3}

As the centre of \mathrm{C_3}, lies above the x-axis, we take \mathrm{\lambda=-\sqrt{3}} and thus an equation of \mathrm{C_3 \: \: is \: \: x^2+y^2-x-\sqrt{3} y=0}

Since \mathrm{C_1 \: \: and \: \: C_3} intersect and are of unit radius, their common tangents are parallel to the line joining their centres \mathrm{(0,0)\: and \: \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)}
So, let the equation of common tangent be \mathrm{\sqrt{3} x-y+k=0}
It will touch \mathrm{C_1}, if  \mathrm{\left|\frac{k}{\sqrt{3+1}}\right|=1 \quad \Rightarrow k= \pm 2}
Above the figure, we observe that the required tangent makes positive intercept on the y-axis and negative on the x-axis and hence its equation is \mathrm{\sqrt{3} x-y+2=0}

Hence option 2 is correct.

Posted by

Irshad Anwar

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

JEE Main 2026: NTA reopens Session 2 registration; apply by March 13 at jeemain.nta.nic.in
anam-khan

JEE Mains 2026 LIVE: NTA session 2 city slip soon; admit card, exam dates, here's how to prepare
anam-khan

JEE Main 2026 session 2 application correction begins on jeemain.nta.nic.in; edit details by tomorrow

Latest Question