1. Line $l$ has equation $y = 4x - 7$, and line $m$ with equation $y = ax + b$ is perpendicular to line $l$ at $(2,1)$. What is the $y$-coordinate of the point on $m$ that has $x$-coordinate 6?

2. Find $B - A$ if the graph of $Ax + By = 7$ passes through $(2,1)$ and is parallel to the graph of $2x - 7y = 3$.

3.The "perpendicular bisector" of the line segment $\overline{AB}$ is the line that passes through the midpoint of $\overline{AB}$ and is perpendicular to $\overline{AB}$. The equation of the perpendicular bisector of the line segment joining the points $(1,2)$ and $(-5,12)$ is $y = mx + b$. Find $m+b$.

Guest Dec 4, 2017

#1**+1 **

1. Line $l$ has equation $y = 4x - 7$, and line $m$ with equation $y = ax + b$ is perpendicular to line $l$ at $(2,1)$. What is the $y$-coordinate of the point on $m$ that has $x$-coordinate 6?

The slope of the perpedicular line is - (1/4)

So...the equation of line m is

y = (-1/4)(x -2) + 1

y = (-1/4)x + 1/2 + 1

y = (-1/4)x + 3/2

So on this line....when x = 6

y = (-1/4)(6) + 3/2

y = -6/4 + 3/0

y = -3/2 + 3/2

y = 0

CPhill
Dec 4, 2017

#2**+1 **

2. Find $B - A$ if the graph of $Ax + By = 7$ passes through $(2,1)$ and is parallel to the graph of $2x - 7y = 3$.

The slope of the first line = -A/B = the slope of the second line = 2/7

Which implies that -7A = 2B ⇒ B = (-7/2)A

So

A(2) + B(1) = 7

A(2) + (-7/2)A = 7

-(3/2)A = 7

A = -14/3

B = (-7/2)(-14/3) = 98/6 = 49/3

So.......the equation of the first line is

(-14/3)x + (49/3)y = 7

And B - A = 49/3 + 14/3 = 63 / 3 = 21

Here's the graph showing this : https://www.desmos.com/calculator/qrhzf2f3op

CPhill
Dec 4, 2017

#3**+1 **

3.The "perpendicular bisector" of the line segment $\overline{AB}$ is the line that passes through the midpoint of $\overline{AB}$ and is perpendicular to $\overline{AB}$. The equation of the perpendicular bisector of the line segment joining the points $(1,2)$ and $(-5,12)$ is $y = mx + b$. Find m + b

The midpoint of AB is [ (1 - 5) / 2, (12 + 2) / 2 = ( - 4/2 14/2) = (-2, 7)

And the slope of AB = [ 12 - 2] / [ -5 - 1 ] = 10 / -6 = - 5/3

So....the perpedicular bisector will have the slope 3/5

And the equation of this bisector is

y = (3/5)(x - - 2) + 7

y = (3/5)x + 6/5 + 7

y = (3/5)x + 41/5

So.....m + b = 3/5 + 41/5 = 44/5

CPhill
Dec 4, 2017