Answers

  • Brainly User
2015-12-01T13:33:12+08:00
Steps in deriving the equation of circle given the endpoints of its diameter:
Given: (4,6) and 0,-2)

1) Find the center (h, k)  using midpoint formula.
Midpoint = ( \frac{x_{1}+x_{2}  }{2},  \frac{y_{1}+y_{2}  }{2}  )

x₁ = 4     x₂ = 0
y₁ = 6     y₂ = -2

Midpoint (Center) = ( \frac{4+0}{2}, \frac{6+ (-2)}{2})
                             = (⁴/₂, ⁴/₂)
                             = (2, 2)

The center (h,k) is (2, 2).

2)  Find the distance of the radius by solving for the distance of the two endpoints of diameter divided by 2.  (Radius is 1/2 of diameter of the circle.)

Radius = ( \sqrt{(x_{2}-x_{1} )^{2} + (y_{2} -y_{1} ) ^{2}) /2

Radius = ( \sqrt{(0-4) ^{2}+(-2-6) ^{2}  })/2

Radius = ( \sqrt{(4) ^{2}+(-8) ^{2}  })/2

Radius = (1/2) \sqrt{(16)(5)}

Radius = (1/2)(4)  \sqrt{5}

Radius = 2 \sqrt{5}

3)  Equation:
Standard or Center-Radius Form:
(x - h)² + (y-h)² = r²

(x - 2)² + (y - 2)² = ((2 \sqrt{5} ) ^{2}

(x-2)² + (y-2)² = (4)(5)

(x-2)² + (y-2)² = 20

4.) Equation of the circle in general form, x² + y² + Cx + Dy + E = 0:
(x-2)² + (y-2)² = 20

x² - 4x + 4 + y² - 4y + 4 - 20 = 0

x² + y² - 4x - 4y - 16 = 0






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