Answer:
a) y = 2x -2
b) Equation: q = 2p - 2 ----(1)
p² + q² = 4 ----(2)
The coordinates of C may be (0, -2) or (6/5, 8/5)
Step-by-step explanation:
a) Step 1: Find the coordinates of midpoint, P of AB
x-co-ordinate of P = (x₁ + x₂)/2 = (-1 + 7)/2 = 3
y- co-ordinate of P = (y₁ + y₂)/2 = (6 + 2)/2 = 4
Step 2: Determine the gradient, m₂ of perpendicular bisector of AB
For perpendicular lines, m₁ * m₂ = -1 where m₁ is slope of AB
m₁ = (2 - 6)/(7 - -1) = -4/8 = -1/2
-1/2 * m₂ = -1
m₂ = 2
Step 3: Deriving the equation of the perpendicular bisector in the form y = mx + c
y - y₁ = m(x - x₁); substituting the values of the midpoints of the line
y - 4 = 2(x - 3)
y - 4 = 2x - 6
y = 2x -2
b) Step 1: Derive the values of the p and using the formula of distance between two points:
y = 2x - 2
therefore q = 2p - 2 ----(1)
OC = √(x₂ -x₁)² + (y₂ - y₁)²
2 = √(p - 0)² + ( q - 0)²
2 = √p² + q²
squaring both sides
p² + q² = 4 ----(2)
Substituting (1) in (2)
p² + (2p - 2)² = 4
p² + 4p² - 8p + 4 = 4
5p² - 8p = 0
p(5p - 8) = 0
Therefore p = 0 or 5p - 8 = 0
p = 0 or p = 8/5
Substituting p = 0 in (1)
q = -2
substituting p = 8/5 in (1)
q = 2(8/5) - 2
q = 16/5 - 2
q = 6/5
Hence the coordinates of C may be (0, -2) or (6/5, 8/5)
