Final answer:
To find q in terms of s, we utilize the section formula and the given ratio in which B divides AC internally. By applying the formula to the coordinates of B in relation to A and C, we can set up an equation and solve for q in terms of s after eliminating the variable r.
Step-by-step explanation:
To express q in terms of s given that points A(4r,2r), B(q,s), and C(4q,6s) lie on a straight line and B divides AC in the ratio 3:4, we can use the concept of section formula in coordinate geometry. In a section formula, if a point P(x1, y1) divides the line segment connecting A(x2, y2) and B(x3, y3) internally in the ratio m:n, then the coordinates of P can be represented as P((mx3 + nx2)/(m+n), (my3 + ny2)/(m+n)).
Applying this to our points, we have B dividing AC in the ratio 3:4, hence the coordinates of B would be B((3*4q + 4*4r)/(3+4), (3*6s + 4*2r)/(3+4)). Since we know that the x-coordinate of B is q and the y-coordinate is s, we can equate them to the respective formulas obtained from section formula and solve for q.
For the x-coordinate: q = (3*4q + 16r)/7. To find q in terms of s, we need an equation involving both these variables. From the y-coordinate of B we get: s = (18s + 8r)/7. By solving simultaneously, we would eliminate r and find an expression for q in terms of s.