Answer:
(0, 4) and (- 4, 0)
Step-by-step explanation:
We require the equation of the circle
The equation of a circle centred at the origin is
x² + y² = r² ( r is the radius )
The radius is the distance from the centre to a point on the circle
To calculate r use the distance formula
r = √ (x₂ - x₁ )² + (y₂ - y₁ )²
with (x₁, y₁ ) = (0, 0) and (x₂, y₂ ) = ([tex]\sqrt{12}[/tex], - 2)
r = [tex]\sqrt{((\sqrt{12 )^2 - 0)+(-2-0)^2} }[/tex]
= [tex]\sqrt{12+4}[/tex] = [tex]\sqrt{16}[/tex] = 4, hence
x² + y² = 16 ← equation of circle
given y - x = 4 ⇒ y = x + 4
Substitute y = x + 4 into the equation of the circle
x² + (x + 4)² = 16 → distribute and simplify left side
x² + x² + 8x + 16 = 16
2x² + 8x + 16 = 16 ( subtract 16 from both sides )
2x² + 8x = 0
2x(x + 4) = 0
Equate each factor to zero and solve for x
2x = 0 ⇒ x = 0
x + 4 = 0 ⇒ x = - 4
Substitute these values into y = x + 4 for corresponding y- coordinates
x = 0 : y = 4 ⇒ (0, 4)
x = - 4 : y = - 4 + 4 = 0 ⇒ (- 4, 0)
The points of intersection are (0, 4) and (- 4, 0)
