A line segment is bisected by a line with the equation # 4 y + 9 x = 8 #. If one end of the line segment is at #(5 ,2 )#, where is the other end?

Answer 1

The other end is at #(1295/97, 554/97 )#

In slope-intercept form, write the following line:

#y = -9/4x + 2#
Because bisector is perpendicular, the slope of the line segment will be the negative reciprocal of its bisector, #4/9#.

The equation for the distance between the point and the line is provided by this reference. It is as follows:

#d = |(4(2) + 9(5) - 8)/(sqrt(4^2 + 9^2))| = 45sqrt(97)/97#
The length of the line segment is twice this distance, #90sqrt(97)/97#.
From point #(5,2)#, we move to the right a distance, (x), and up a distance y , we know that y is #4/9x#, and we know that length of the hypotenuse formed by this right triangle is #90sqrt(97)/97#
#(90sqrt(97)/97)^2 = x^2 + (4/9x)^2#
#90^2/97 = 81/81x^+ 16/81x^2#
#x^2 = 81(90^2)/97^2#
#x = 810/97#
#y = 360/97#

Point is where the line's other end is located.

#(5 + 810/97, 2 + 360/97) = (1295/97, 554/97 )#
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Answer 2

First, find the slope of the given line by rearranging its equation into slope-intercept form (y = mx + b):

[ 4y + 9x = 8 ] [ 4y = -9x + 8 ] [ y = -\frac{9}{4}x + 2 ]

The slope of the line is ( m = -\frac{9}{4} ).

Since the given line bisects the line segment, the slope of the line segment is the negative reciprocal of ( -\frac{9}{4} ), which is ( \frac{4}{9} ).

Using the point-slope form of a line (y - y1 = m(x - x1)), where (x1, y1) is the given point (5, 2) and m is the slope ( \frac{4}{9} ):

[ y - 2 = \frac{4}{9}(x - 5) ]

Now, solve for y:

[ y = \frac{4}{9}x - \frac{20}{9} + 2 ] [ y = \frac{4}{9}x - \frac{20}{9} + \frac{18}{9} ] [ y = \frac{4}{9}x - \frac{2}{9} ]

So, the other end of the line segment is at the point ( \left( x, \frac{4}{9}x - \frac{2}{9} \right) ), where x can take any real value.

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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.

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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