The area (in sq units) of the equilateral triangle formed by the tangent at ( 3 , 0) to the hyperbola x^2-3 y^2=3 with the pair of asymptotes of the hyperbola is

Step-by-step solution

Given equation of hyperbola is x^2-3 y^2=3 Equation of tangent at point ( 3 , 0) is aligned & S₁=0 & x 3 -3 y 0=3 aligned aligned & & x 3 & =3 & & x & = 3 aligned The asympotes of given hyperbolas are x+ 3 y=0 and x- 3 y=0 On solving Eqs. (i), (ii) and (iii), we get (0,0),( 3 ,-1) and ( 3 , 1) Area of triangle formed by joining the above points aligned & = 1 2 | array rrr 0 & 0 & 1 3 & -1 & 0 3 & 1 & 0 array | & = 1 2 [1( 3 + 3 )]= 1 2 2 3 = 3 aligned