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Find the reflection of the point (10,9) in the line 4x - 3y + 12 = 0.

 Jan 20, 2021
 #1
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4x   -3y  +  12  = 0     rearrange as

 

3y = 4x  + 12

 

y  = (4/3)x  +  4   (1)

 

A perpendicular   through   (10, 9)   will  have a negative reciprocal slope  =  -3/4

 

The equation of this line is 

y= (-3/4) ( x  -10) + 9

y = (-3/4)x  + 30/4 + 9

y =  (-3/4)x + 33/2     (2)

 

Find  the x coordinate of the intersection of (1)  and (2)

 

(4/3)x + 4 =  (-3/4)x  + 33/2

(4/3 + 3/4)x  =  33/2  - 4

( 25/12)x  = 25/2

x = ( 25/2) ( 12/25)  =  12/2  = 6

And y  = (4/3)(6) + 4 =  12

(6, 12)

 

This is the  midpoint  between    (9,10)  amd  the reflection point

 

So we have

 

(10 + x)/2  = 6                   (9 + y)  / 2  =  12

10 + x  = 12                         9 + y  =  24

x = 2                                    y = 15

 

The reflected point  is  (2,15)

 

Here's a graph  :  https://www.desmos.com/calculator/l4wolqqhos

 

 

cool cool cool

 Jan 20, 2021

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