In the figure below OQ=q and OR=r.Point X divides OQ in the ratio 1:2 and Y divides OR in the ratio 3:4 lines XR and YQ interest at E If XE=mXR and YE=nYQ , express OE in terms of r,q andm n,r,q?

To find the ratio in which point E divides line OR, we can use the properties of similar triangles.

Let's denote the length of OX as x and OY as y. Since X divides OQ in the ratio 1:2, we have:

x = (1/3)q
OX = x + OQ = (1/3)q + q = (4/3)q

Similarly, since Y divides OR in the ratio 3:4, we have:

y = (3/7)r
OY = y + OR = (3/7)r + r = (10/7)r

Now, we can find the coordinates of points X and Y:

X = (4/3)q * (1/3)q = (4/9)q
Y = (10/7)r * (3/7)r = (30/49)r

Now, we can find the coordinates of point E by using the section formula:

E = ((m*X + XQ)/(m+1), (n*Y + YR)/(n+1))

Substitute the values of X, Y, XQ, and YR into the formula:

E = ((m*(4/9)q + (1/3)q)/(m+1), (n*(30/49)r + (3/7)r)/(n+1))

E = ((4m/27 + 1/3)q/(m+1), (30n/49 + 3/7)r/(n+1))

Now, we can express OE in terms of r, q, m, and n:

OE = sqrt((x - Ox)^2 + (y - Oy)^2)

OE = sqrt(((4m/27 + 1/3)q/(m+1) - q)^2 + ((30n/49 + 3/7)r/(n+1) - r)^2)

Therefore, OE can be expressed in terms of r, q, m, and n.