If the equation of a parabola, whose focus is at (-1, 7) and the tangent at its vertex is 4x - 3y = 0 , is given by 9x^2 + ay^2 + bxy + cx + dy + k = 0 , then the value of a + b + c + d + k is equal to

Step-by-step solution

Given the focus S(-1, 7) and the tangent at the vertex 4x - 3y = 0 . The foot of the perpendicular from the focus to the tangent at the vertex is the vertex V of the parabola. Let V be (x₁, y₁) . Using the foot of perpendicular formula: x₁ - (-1) 4 = y₁ - 7 -3 = - 4(-1) - 3(7) 4^2 + (-3)^2 = - -25 25 = 1 x₁ + 1 = 4 x₁ = 3 y₁ - 7 = -3 y₁ = 4 So, the vertex is V(3, 4) . Let Z be the point of intersection of the axis and the directrix. The vertex V is the midpoint of S and Z . Z = 2V - S = (2(3) - (-1), 2(4) - 7) = (7