Question

If distance between two parallel tangents having slope m drawn to the hyperbola x29−y249=1 is 2,x29−y249=1 is 2, is:

Answer 1

Ans Given, $\frac{x^2}{9}-\frac{y^2}{49}=1$

we know that, the equation of

hyperbola is of form

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

also,

$c^2=a^2{m}^2-b^2$

so, $c^2=9m^2-49$

$c=\pm \sqrt{9m^2-49}$

equation of line is $y=mx+c$

$y=mx\pm \sqrt{9m^2-49}$

$x+y=c_1$ $x+y=c_2$

Distance $=\left|\frac{c_1-c_2}{(coffi.ofx{)}^2+(coffi.ofy{)}^2}\right|$

$2=\left|\frac{2\sqrt{9m^2-49}}{\sqrt{1+m^2}}\right|$

$\frac{\sqrt{9m^2-49}}{1+m^2}=\pm 1\Rightarrow \frac{9m^2-49}{1+m^2}=1$

$\Rightarrow 9m^2-49=m^2+1\Rightarrow 8m^2=50$

$\Rightarrow m^2=\frac{50}{8}$ $m^2=\frac{25}{4}$ $m=\pm \frac{5}{2}$

$\Rightarrow 2\left|m\right|=2\times \frac{5}{2}$ $2|m|=5$