If one focus of a hyperbola is ( 3 , 0 ) , the equation of its directrix is 4 x - 3 y - 3 = 0 and its eccentricity e = 5 4 , then the co-ordinates of its vertex is

Step-by-step solution

Distance from focus to directrix is a - 1 e + e = a - 4 5 + 5 4 = 9 a 20 Perpendicular distance of focus from directrix = 4 × 3 - 3 × 0 - 3 4 2 + 3 2 ⇒ 9 a 20 = 9 5 ⇒ a = 4 The slope of the axis is - 3 4 , since the slope of the directrix is 4 3 Traveling along the axis, parametric equations of point are 3 + r × - 4 5 , 0 + r 3 5 r = a e - a = 4 × 5 4 - 1 = 1 Vertex = 3 - 4 5 , 3 5 = 11 5 , 3 5