# A line segment is bisected by a line with the equation  - 9 y + 4 x = 6 . If one end of the line segment is at ( 7 , 1 ), where is the other end?

May 27, 2016

The coordinate of any point on the straight line represented by equation $4 x - 9 y + 7 = 0$ ,will be the coordinate of other end point.

#### Explanation:

The coordinate of one end of the given line segment is (7,1).
Let the coordinate of other end be (h,k).

Hence the coordinate of its middle point is $\left(\frac{h + 7}{2} , \frac{k + 1}{2}\right)$

The given line having equation $4 x - 9 y = 6. \ldots \ldots . \left(1\right)$ bisects the line segment.

So the mid point of the line segment is lying on the given straight line.
Hence its coordinate will satisfy the given equation.

So inserting $\left(x = \frac{h + 7}{2} \mathmr{and} y = \frac{k + 1}{2}\right)$ in the given equation we have

$4 \times \frac{h + 7}{2} - 9 \times \frac{k + 1}{2} = 6$

$\implies 4 h + 28 - 9 k - 9 = 12$

$\implies 4 h - 9 k + 7 = 0$

Inserting x for h and y for k we get an equation of the straight line

as $4 x - 9 y + 7 = 0. \ldots . . \left(2\right)$ which is a straight line parallel to the straight line represented by equation (1)

Hence the coordinate of any point on the straight line represented by equation $4 x - 9 y + 7 = 0. \ldots . . \left(2\right)$ will be the coordinate of other end point. 