The point A has coordinates (1, 1) and the point B has coordinates (5, k). The line AB has equation 3x+4y=7.

a. (i) show that k=2

(ii) hence find the coordinates of the mid-point of AB

b. (i) find the gradient of AB

c. The line AC is perpendicular to the line AB.

(i) Find the gradient of AC.

(ii) Hence find an equation of the line AC.

(iii)Given that the point C lies on the x-axis, find its x-coordinate.

Part II

The line L has equation y+2x=12 and the curve C has equation y=x^{2} -4x +9.

(i) Show that the x-coordinates of the points of intersection of L and C satisfy the equation x^{2} - 2x - 3 = 0

(ii) Hence find the coordinates of the points of intersection of L and C