Question

A line segment is bisected by a line with the equation `2 y - 5 x = 2`. If one end of the line segment is at `( 4 , 8 )`, where is the other end?

Answer 1

`color(magenta)("Coordinates of the other end point " D (24/29, 176/29)`

Explanation:

Let AB be the line which is a perpendicular of segment CD.

Equation of line AB `2y - 5x = 2 " Eqn (1)"`

`C (4,8)`

`y = (5/2)x + 1`

`"Slope of line AB " m_1 = 5/2`

Slope of segment CD `m_2 = -1/ m_1 = -2/5`

Equation of line segment CD " (y - 8) = -(2/5) (x - 4)#

`5y - 40 = -2x + 8`

`5y + 2x = 48, " Eqn (2)`

Solving Eqns (1), (2), we get the intersection point E as `(70/29, 204/29)`

Let #x_d, y_d " be the coordinates of point D. Then,

`x_d = (2 * x_e) - x_c = 140 / 29 - 4 = 24/29`

`y_d = (2 * y_e) - y_c = 408/29 - 8 = 176/29`