Question

Area of the triangle formed by the lines x-y=0, x+y=0 and any tangent to the hyperbola
$x^2-y^2=a^2$ is

• A. $|a|$
• B. $\frac{1}{2}|a|$
• C. $a^2$
• D. $\frac{1}{2}{a}^2$

Answer 1

The correct option is A

$|a|$

$\because$ x-y =0 and x+y=0
are the asymptotes of the rectangular hyperbola $x^2-y^2=a^2$
Equation of tangent at $P(asec\varphi ,atan\varphi )$ of $x^2-y^2=a^2$ is

$axsec\varphi -aytan\varphi =a^2\text{or}xsec\varphi -ytan\varphi =a...........\left(i\right)\text{Solving}y=x\text{and}y=-x\text{with Eq.~(i),then we get}A\left(a\right(sec\varphi +tan\varphi ),a(sec\varphi +tan\varphi \left)\right)B\left(a\right(sec\varphi -tan\varphi ),a(tan\varphi -sec\varphi \left)\right)\therefore \text{Area of}\Delta CAB=\frac{1}{2}\left|a\right(ta{n}^2\varphi -se{c}^2\varphi )-a(se{c}^2\varphi -ta{n}^2\varphi \left)\right|=\frac{1}{2}|-a-a|=|-a|=\left|a\right|$