The equation of perpendicular bisector of A(-6, -4 ) and B(2, 0) is y = -2x - 6
Solution:
Given that we have to find the equation of perpendicular bisector of A(-6, -4 ) and B(2, 0)
A perpendicular bisector, bisects a line segment at right angles
To obtain the equation we require slope and a point on it
Find the midpoint and slope of the given points and then we can find the equation
Find the midpoint:
Given points are A(-6, -4 ) and B(2, 0)
The midpoint is given as:
[tex]m(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)[/tex]
[tex]\text {Here } x_{1}=-6 ; y_{1}=-4 ; x_{2}=2 ; y_{2}=0[/tex]
Substituting the values we get,
[tex]\begin{aligned}&m(x, y)=\left(\frac{-6+2}{2}, \frac{-4+0}{2}\right)\\\\&m(x, y)=(-2,-2)\end{aligned}[/tex]
Find the slope of given points:
[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
[tex]m=\frac{0-(-4)}{2-(-6)}\\\\m = \frac{4}{8}\\\\m = \frac{1}{2}[/tex]
Then the slope of perpendicular bisector is given as:
We know that product of slopes of given line and slope of line perpendicular to it is equal to -1
Let the slope of perpendicular bisector be [tex]m_1[/tex]
[tex]\frac{1}{2} \times m_1 = -1\\\\m_1 = -2[/tex]
Find the equation of line with slope -2 and point (-2, -2)
The equation of line in slope intercept form is given as:
y = mx + c -------- eqn 1
Where "m" is the slope and "c" is the y - intercept
Substitute (x, y) = (-2, -2) and slope m = -2 in eqn 1
-2 = -2(-2) + c
-2 = 4 + c
c = -2 - 4
c = -6
Substitute c = -6 and m = -2 in eqn 1
y = -2x - 6
Thus the required equation of perpendicular bisector is found