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Hi, If you help me it basically means you saved my life!

 

1. Find the distance (3,4) between  and the line 4x+3y+7=0.

 

2. Find the distance between the graphs of 5x-12y+45=0 and 5x-12y-46=0.

 

3. A circle centered at the origin is tangent to the line x-y+4=0. What is the area of the circle?

 

THANK YOU!

Ashra  May 12, 2018
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4+0 Answers

 #1
avatar+86613 
+1

1.

 

There may be several  ways to do this, but one of the easier ways  is through  the "formula" for finding the distance between a point and aline

 

If  (m, n)  is the point  and  Ax + BY + C = 0   is the equation of the line....the distance is given by

 

l  Am + Bn + C  l              l  4(3) + 3(4) + 7  l              l  31  l               31

______________  =  __________________  =    _______      =       ___  =  6.2    units

 √ [ A^2 + B^2 ]              √ [ 4^2 + 3^2  ]                    √  25                   5

 

 

 

cool cool cool 

CPhill  May 12, 2018
 #2
avatar+86613 
+1

2.

 

5x-12y+45=0   and  5x-12y-46=0

 

12y   = 5x  + 45     and      12y  = 5x  - 46

 

y  = (5/12)x + 45/12

 

y = (5/12) x  +  15/4

 

 

These lines have the same slope.......the first line has  a  y intercept at  15/4....

 

 

So the point  (0, 15/4)  is on the first line

 

The distance between this point and the second line is the perpendicular distance between both lines...so we have

 

l  5(0)  - 12 (15/4) - 46  l               l  -12 (15/4)  - 46  l            l  -91  l              91       

____________________    =  ________________  =          ______     =      ___    =   7 units

      √ [ 5^2  + 12^2  ]                      √ [ 169 ]                             13                   13

 

 

cool cool cool

CPhill  May 12, 2018
 #3
avatar+86613 
+1

3. A circle centered at the origin is tangent to the line x-y+4=0. What is the area of the circle?

 

 

Since the circle is centered at  (0,0)  a radius drawn  from this point will be perpendicular to this line at the point of tangency....and this will be the shortest distance from (0,0) to this line....so.....this distance  is given by

 

l  1 (0) -  1 (0)   + 4  l                       4          4√2             2√2     =  √8  units  = radius  of the circle

________________       =            ___  =    ____   =

   √  [1^2  + 1^2 l                            √2            2

 

So....the area  of the circle  = pi * radius ^2    =    pi  *  ( √8)^2   =   8 pi units^2

 

 

 

cool cool cool

CPhill  May 12, 2018
 #4
avatar+7 
0

THANK YOU SO MUCH!

Ashra  May 14, 2018

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