Question

Tangents drawn from $(b,a)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ make angles ${\theta }_1,{\theta }_2$ with x-axis. If $\tan {\theta }_1\tan {\theta }_2=2$ then $b^2-a^2=$

• A. $a^2+2b^2$
• B. $2a^2+b^2$
• C. $\frac{a^2+b^2}{2}$
• D. $a^2-b^2$

Answer 1

Answer: (C)

$\frac{a^2+b^2}{2}$

Equation of line through $(b,a)$ is
$y-a=m(x-b)$
$y=mx+(a-mb)$
Condition for line $y=mx+c$ to be a tangent to hyperbola is $c^2=a^2{m}^2-b^2$
$\Rightarrow (a-mb{)}^2=a^2{m}^2-b^2$
$\Rightarrow a^2+m^2{b}^2-2amb=a^2{m}^2-b^2$
$m^2(b^2-a^2)-2abm+a^2+b^2=0$
$\Rightarrow m_1{m}_2=tan{\theta }_{1}tan{\theta }_2=\frac{a^2+b^2}{b^2-a^2}=2$
$\Rightarrow b^2-a^2=\frac{a^2+b^2}{2}$