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# help coordinates

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Find the shortest distance from the line 3x + 4y = 25 to the circle x^2 + y^2 = 6x - 8y.

Jun 8, 2022

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First.....we need to find the circle center

Rewrite as

x^2 - 6x   + y^2 + 8y   = 0              complete the square on  x and y

x^2 - 6x + 9  + y^2 + 8y + 16 =   9 + 16

(x - 3)^2  +  ( y + 4)^2  = 25

This is a circle with a center of ( 3, -4)     and  a radius of  5

Now.....we can use  a special formula that will tell us how  far the  center of the circle is  from our given line

We are using the equation of the line in standard  form and  filling in x  and  y with the circle's center

The squre root in the  denominator is just the coefficients of the  line (each squared)

We have that

l  3(3) + 4(-4) - 25  l                 l -32 l              32

________________   =       ________  =      __

sqrt  [ 3^2 + 4^2 ]                 sqrt (25)            5

Subtract the radius of the circle from this

32 / 5    -  5   =

32 /5   -   25 /5   =

7/5  =  1.4   units  = the shortest distance  between the given line  and  the circle

Jun 8, 2022