A line segment is bisected by a line with the equation 4 y - 3 x = 2 . If one end of the line segment is at ( 7 , 5 ), where is the other end?

2 Answers
Jul 7, 2016

The other point is on the line color(green)(4y-3x=5)
(any point on this line will satisfy the given requirement)

Explanation:

Given
color(white)("XXX")The line color(red)(4y-3x=2)
and a point color(purple)(""(7,5))

Consider the vertical line through color(purple)(""(7,5))
This vertical line will intersect color(red)(4y-3x=2) at color(red)(""(x=color(purple)(7),y))
where color(red)(y) can be determined by solving
color(white)("XXX")color(red)(4y-3xxcolor(purple)(7)=2)

color(white)("XXX")rarr color(red)(y=23/4)

The distance from color(purple)(""(7,5)) to color(red)(""(7,23/4))
is color(red)(23/4)-color(purple)(5)=3/4

So a point color(green)(""(7,5+2xx3/4) = (7,13/2)) will be twice as far away from color(purple)(""(7,5)) as the point color(red)(""(7,23/4)) on the same vertical line

Note that any point on a line parallel to color(red)(4y-3x=2 through color(green)(""(7,13/2)) will also be twice as far away from color(purple)(""(7,5)) as the point from color(purple)(""(7,5)) on color(red)(4y-3x=2) to that point.
enter image source here

If color(red)("L1") parallel to color(green)("L2")
then triangle ABC ~=triangle ADE
rarr abs(AB):abs(AD)=abs(AC):abs(AE)

Since color(red)(4y-3x=2) has a slope of color(red)(3/4)

Our required line will also have a slope of color(green)(3/4)
and since it passes through (""(7,13/2))
using the slope-point form, we have
color(white)("XXX")color(green)(y-13/2=3/4(x-7))
or
color(white)("XXX")color(green)4y-3x=5
enter image source here

Jul 7, 2016

The other end-pt. lies on the line given by the eqn. 4y-3x=5.

Explanation:

Suppose that the other end-pt. is P(X,Y).

Let the given end-pt. be Q(7,5), and the given line be L : 4y-3x=2.

If M is the mid-pt. of the segment PQ, then co-ords. of M as obtained by using Section Formula for Mid-pt. are =M((7+X)/2,(5+Y)/2)

Now, PQ is bisected by L at M, so, M in L.

Therefore, co-ords. of M must satisfy the eqn. of L.

Hence, 4{(5+Y)/2}-3{(7+X)/2}=2 rArr 20+4Y-21-3X=4, i.e., 4Y-3X=5, as Sir Alan P. has readily derived!

This shows that :
(i) the co-ords. of other end-pt. can not be uniquely derived under the given conds.
(ii) What we can say about it (the other end-pt.) is that it lies on : 4y-3x=5. This eqn. represents a line || to L.