Answer:
a) -2
b) y = -2x + 6
Step-by-step explanation:
Question (a)
Point A = (-1, -2)
Point B = (7, 2)
[tex]\sf gradient\:of\:AB=\dfrac{change\:in\:y}{change\:in\:x}=\dfrac{2-(-2)}{7-(-1)}=\dfrac{4}{8}=\dfrac12[/tex]
If lines are perpendicular to each other, the product of their gradients will be -1.
Let g be the gradient of the perpendicular line:
[tex]\implies \sf \dfrac12 \times g=-1[/tex]
[tex]\implies \sf g=-1 \div \dfrac12=-2[/tex]
Therefore, the gradient of the line perpendicular to AB is -2
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Question (b)
[tex]\sf midpoint\:of\:line = \left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)[/tex]
[tex]\sf let\:point\:A=(x_1,y_1)=(-1,-2)[/tex]
[tex]\sf let\:point\:B=(x_2,y_2)=(7,2)[/tex]
[tex]\sf \implies midpoint\:of\:line\:AB = \left(\dfrac{-1+7}{2},\dfrac{-2+2}{2}\right)=(3,0)[/tex]
We now know the gradient and a point on the perpendicular line.
So we can use the point-slope form of a linear equation:
[tex]\sf y-y_1=m(x-x_1)[/tex]
[tex]\implies \sf y-0=-2(x-3)[/tex]
[tex]\implies \sf y=-2x+6[/tex]
