The coordinates of the midpoint of the line AB is (0, 1/2).
The equation of the curve is given by y = 2xy + 5.
The equation of the line is given by 2x + 5y = 1.
We have to first find the point of intersection between the curve and the line and then find the coordinates of the midpoint of line AB where A and B is the point of intersection between the curve and the line.
What is the formula for finding the coordinates of a point that divides a given line in the ratio m:n?
If C(x, y) divides a line AB in m:n then we have,
[tex]x = \frac{mx_2 +nx_1}{m+n},~~~y =\frac{my_2+ny_1}{m+n}[/tex]
And if C(x,y) is the midpoint then m = n.
we have,
[tex]x = \frac{x_2 +x_1}{2},~~~y =\frac{y_2+y_1}{2}[/tex]
Where x and y are the coordinates of the midpoint o the line AB.
Find the point of intersection between the line and the curve.
y = 2xy + 5............(1)
2x + 5y = 1..............(2)
From (2) we have,
2x = 1 - 5yx = (1-5y) / 2..........(3)
Substituting (3) in (1).
[tex]y =2\frac{(1-5y)}{2}y + 5\\\\y =\frac{2y-10y^2}{2} + 5\\\\2y = 2y - 10y^2 + 10\\\\10y^2 = 10\\\\y^2 = 1[/tex]
So we have,
y = 1 and y = -1
Puttin y = 1 in (3) we get,
x = (1 - 5) / 2 = - 4 / 2 = -2
Puttin y = -1 in (3) we get,
x = {1-5(-1)} / 2 = (1+5) / 2 = 6 / 2 = 3
Now we have two points of intersection A( 1, -2 ) and B( -1, 3 ).
Finding the coordinates of the midpoint of the line AB.
we have,
[tex]A(1, -2) = A (x_1, y_1)~~and~~B(-1, 3) = B(x_2, y_2)[/tex]
Substituting in the given equation.
[tex]x = \frac{x_2 +x_1}{2},~~~y =\frac{y_2+y_1}{2}\\\\x = \frac{-1 +1}{2},~~~y =\frac{3+(-2)}{2}\\\\x = 0,~~~y=\frac{1}{2}[/tex]
So the coordinates of the midpoint is (x,y) = ( 0, 1/2 ).
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