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A line segment is bisected by a line with the equation # 2 y -4 x = 1 #. If one end of the line segment is at #(3 ,8 )#, where is the other end?

Answer 1

The other end of the line segment is (4.2, 7.4)

Let the other end of the line segment be #(x_1,y_1)# So the mid-point of #(3,8)# and #(x_1,y_1)# is #((x_1+3)/2,(y_1+8)/2)# The mid point belongs to the line #2y-4x=1# The slope of this line is#=2#

Sustituting the values of the mid point in this equation #(2(y_1+8))/2-4((x_1+3))/2=1# Simplifying the equation #y_1+8-2(x_1+3)=1# #y_1+8-2x_1-6=1# #y_1-2x_1=1-8+6=-1#

We need the equation of the line segment the slope #m=-1/2# since the two lines are perpendicular and the product of the slopes is #m_1*m_2=-1#

so the equation is #(y_1-8)/(x_1-3)=-1/2# #2y_1+x_1=19# #y_1-2x_1=-1# Solving we get #(y_1=2x_1-1)# Substituting in other equation #2(2x_1-1)+x_1=19# #4x_1+x_1=21# #=>##x_1=21/5=4.2# and #y_1=-1+2*4.2=7.4#

The key point is thus (4.2,7.4)

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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.