A line segment is bisected by a line with the equation 4 y + x = 3 . If one end of the line segment is at (5 ,6 ), where is the other end?

1 Answer
Douglas K.
Feb 25, 2018

The other end is at (33/17,-106/17)

Explanation:

Given the equation of the perpendicular bisector of the form:

Ax+By = C_1

The equation of the bisected line is of the form:

Bx-Ay=C_2

Writing the equation of the given perpendicular bisector in the above form:

x+4y = 3

Now, we can with the general form for the bisected line:

4x-y = C_2

To find the value of C_2, we substitute the point (5,6):

4(5)-6 = C_2

C_2 = 14

The equation of the bisected line is:

4x-y = 14

Find the point of intersection of the lines:

x+4y = 3
4x-y = 14

Multiply the second equation by 4 and add to the first equation:

17x= 59

Solve for x:

x = 59/17

Use the given line to solve for y:

59/17+4y = 3

y = -2/17

We can use the midpoint equations to find the other end:

x_"mid" = (x_"start"+x_"end")/2

y_"mid" = (y_"start"+y_"end")/2

Substitute (5,6) for the starts and (59/17,-2/17) for mids:

59/17 = (5+x_"end")/2

-2/17 = (6+y_"end")/2

Solve for (x_"end", y_"end")

x_"end" = 33/17

y_"end" = -106/17

The other end is at (33/17,-106/17)

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