Step 1
Find the equation of the line AB
we know that
the equation of the line into point-slope form is equal to
\(y-y1=m(x-x1)\)
Let
\(A( -2,6)\\B(2,-2)\)
Find the slope m
we know that
the slope between two points is equal to
\(m=\frac{(y2-y1)}{(x2-x1)}\)
substitute
\(m=\frac{(-2-6)}{(2+2)}\)
\(m=\frac{(-8)}{(4)}\)
\(m=-2\)
with the slope m and point A find the equation of the line AB
\(y-6=-2(x+2)\)
or
with the slope m and point B find the equation of the line AB
\(y+2=-2(x-2)\)
we will proceed to verify each case to determine the solution of the problem
Step 2
Verify case A
Case A) \(y-2=-2(x+6)\)
the slope of the line is \(m=-2\)
but the point \((-6,2)\) -----> not lie on the line AB (see the graph)
therefore
the case A is not the solution
Step 3
Verify case B
Case B) \(y-6=-2(x+2)\)
the slope of the line is \(m=-2\)
and the point \((-2,6)\) -----> is the point A
therefore
the case B is a solution
Step 4
Verify case C
Case C) \(y-2=-2(x+2)\)
the slope of the line is \(m=-2\)
but the point \((-2,2)\) -----> not lie on the line AB (see the graph)
therefore
the case C is not the solution
Step 5
Verify case D
Case D) \(y-2=2(x+2)\)
the slope of the line is \(m=2\) -----> the slope is not equal to the slope AB
and the point \((-2,2)\) -----> not lie on the line AB (see the graph)
therefore
the case D is not the solution
the answer is
The equation \(y-6=-2(x+2)\) is a point-slope form of the line AB
