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Answers
Answer:
( -16, -2 )
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Step-by-step explanation:
Here's one approach.
Let A = ( -8, 12 ).
Let B be the image of A.
The line AB is perpendicular to the line 4x + 7y + 13 = 0, so it has equation
7x - 4y + c = 0
for some c.
Since it passes through A,
(7)(-8) - (4)(12) + c = 0 => c = 56 + 48 = 104
So this line is 7x - 4y + 104 = 0.
Let C be the point where the two lines intersect. This will be useful because C is the midpoint of AB. To find C, solve the system of linear equations:
4x + 7y = -13 and 7x - 4y = -104
=> 16x + 28y = -52 and 49x - 28y = -728
=> (16+49)x = -52 - 728
=> 65x = -780
=> x = -12
=> y = ( -13 - 4x ) / 7 = ( -13 + 48 ) / 7 = 35 / 7 = 5
So C = ( -12, 5 )
As C is the midpoint of AB, we have
C = ( A + B ) / 2
=> B = 2C - A = ( -24, 10 ) - ( -8, 12 ) = ( -24 + 8, 10 - 12 ) = ( -16, -2 )