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Question

Only a math genius can solve this


If triangle AOB is given by lines x^2 -4xy + y^2 = 0

and Line AB is Given by 2x+3y=1


Then find the equation of median through O

Answer 1

Equation is: y = (7/8)x

Details later when I have more time.

EDIT:

Line AB is : 2x + 3y = 1, or
(1) y = (-2/3)x + 1/3

Lines AO and BO are given by : x^2 - 4xy + y^2 = 0.
Rearrange slightly to: y^2 - 4xy + x^2 = 0
and solve the quadratic for y.

y = {-(-4x) ± sqrt[(-4x)^2 - (4)(1)(x^2)} / 2

Thus, y = [2 ± sqrt(3)]x

So, let AO be the line:
(2) y = [2 + sqrt(3)]x

and let BO be the line:
(3) y = [2 - sqrt(3)]x

To find point A, we need the intersection of lines AO and AB.
So we equate (2) with (1) to get:

[2 + sqrt(3)]x = (-2/3)x + 1/3

From this, we find that x = [8 - 3*sqrt(3)] / 37,
and on substituting into (2), we get y = [7 + 2*sqrt(3)] / 37.

To find point B, we need the intersection of lines BO and AB.
So we equate (3) with (1) to get:

[2 - sqrt(3)]x = (-2/3)x + 1/3

This gives: x = [8 + 3*sqrt(3)] / 37
and y = [7 - 2*sqrt(3)] / 37.

The median from O to AB bisects line AB, say, at P.
The x-coordinate of P is the average
of the x-coordinates of points A and B.
The y-coordinate of P is the average
of the y-coordinates of points A and B.

x-coordinate of P
= {[8 - 3*sqrt(3)] / 37 + [8 + 3*sqrt(3)] / 37} / 2
= 8 / 37

y-coordinate of P
= {[7 + 2*sqrt(3)] / 37 + [7 - 2*sqrt(3)] / 37} / 2
= 7 / 37

Now we have two points of the median:
O (0,0) because (1) and (2) both intersect the origin,
and P (8/37, 7/37).

The equation of the straight line linking two points is:
(y - y1) / (x - x1) = (y2 - y1) / ( x2 - x1)

Letting (x1, y1) = (0, 0) and (x2, y2) = (8/37, 7/37),
we have:
y / x = (7/37) / (8/37) = 7/8
Therefore, y = (7/8)x is the median line through O.

Answer 2

42. This is also the answer to the question of life.