A line segment is bisected by a line with the equation # -6 y - x = 3 #. If one end of the line segment is at #( 5 , 1 )#, where is the other end?

Answer 1

Coordinates of the other end are #(167/37), -(84/37)#

Assumption : The line bisecting the segment is its perpendicular bisector.

Eqn of line # x + 6y = -3# Eqn (1) #6y = -x -3# #y = -(x/6) - (1/2)# Slope of the line is -(1/6)#
Slope of altitude the line segment is 6 Eqn of line segment is #y -1 = 6 (x-5)# #y - 6x = 1-30 = -29# Eqn (2) Solving Eqns (1) & (2) we get the intersection of the lines or the midpoint of the line segment. Coordinates of mid point are #(171/37), -(47/37)#
#171/37 = (5+x_2)/2, -(47/37) = (1+y_2)/2# where #x_2, y_2 #are the coordinates of the other end point.
#x_2 = (342/37) -5 = 167/37# #y_2 = -(47/37)-1 = -(84/37)#
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Answer 2

#(157/37 , -131/37 )#

First we need to find the equation of the line that is perpendicular to the line #-6y-x=3# and passes through the point #( 5 , 1 )#. Since these lines are perpendicular we can find the gradient of the required line by using the fact that, if #m_1#is the gradient of the known equation, then #m_1*m_2=-1#

#:.#

#-1/6*m_2=-1=>m_2=6#

So second equation is:

#y-1=6(x-5)=>y=6x-29#

The point of intersection of these lines is the coordinates of the midpoint of the line segment.

Solving simultaneously:

#-1/6x-1/2=6x-29=>x=171/37#

#y=6(171/37)-29=-47/37#

We know for a line segment with coordinates #( x_1 , y_2)# and #( 5 , 1 )#, that the coordinates of the midpoint are #((5+x_1)/2 , (1+y_1)/2)#.

Midpoint coordinates:

#(171/37 , -47/37 )#

#:.#

#(5+x_1)/2=171/37=>x_1 =157/37#

#(1+y_1)/2=-47/37=>y_1=-131/37#

Coordinates of end point of line segment:

#(157/37 , -131/37 )#

Plot:

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Answer 3

To find the other end of the line segment bisected by the line with the equation -6y - x = 3, we first need to determine the coordinates of the point where this line intersects the line segment. We'll do this by substituting the coordinates of the given endpoint (5, 1) into the equation of the bisecting line and solving for the other endpoint.

Given endpoint: (5, 1) Equation of bisecting line: -6y - x = 3

Substituting the coordinates of the endpoint into the equation: -6(1) - 5 = 3

Solving for the value of x: -6 - 5 = 3 -11 = 3 x = -11 - 3 x = -14

So, the x-coordinate of the other endpoint is -14. To find the y-coordinate, we substitute the x-value into the equation of the bisecting line.

-6y - (-14) = 3 -6y + 14 = 3 -6y = 3 - 14 -6y = -11 y = -11 / -6 y = 11/6

Therefore, the other end of the line segment is at (-14, 11/6).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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