Question

Can someone help we this and explain how to do it?

Answer 1

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 ) = ($\sqrt{12}$, - 2)

r = $\sqrt{((\sqrt{12 )^2 - 0)+(-2-0)^2} }$

  = $\sqrt{12+4}$ = $\sqrt{16}$ = 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)