Discussion Forum

Tangents are drawn to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ parallel to the straight line 2x-y=1.The points of contacts of the tangents on the hyperbola are

Answer:

Explanation:

Concept lnvolved

Equation of tangent   to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is

  $y= mx \pm \sqrt{a^{2}m^{2}-b^{2}}$

Description of Situation If two straight
lines

  $a_{1}x+b_{1}y+c_{1}=0$

 and $ax_{2}+b_{2}y+c_{2}=0$ are identical

 $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

 sol., Equation of tangent   , parallel to 

 y=2x-1

 $\Rightarrow$   $y=2x\pm\sqrt{9(4)-4}$

 $\therefore$   $y =2x \pm \sqrt{32}$....(i)

 The equation of tangent at $(x_{1},y_{1})$ is 

 $\frac{xx_{1}}{9}-\frac{yy_{1}}{4}=1$ ...(ii)

from Eqs(i) and (ii) , we get 

 $\frac{2}{\frac{x_{1}}{9}}=\frac{-1}{\frac{-y_{1}}{4}}=\frac{\pm\sqrt{32}}{1}$

$\Rightarrow$     $x_{1}=-\frac{9}{2\sqrt{2}} and y_{1}=-\frac{1}{\sqrt{2}}$

 or   $x_{1}=\frac{9}{2\sqrt{2}}$ and  $y_{1}=\frac{1}{\sqrt{2}}$