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Sagot :
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 = [tex]( \frac{x_{1}+x_{2} }{2}, \frac{y_{1}+y_{2} }{2} )[/tex]
x₁ = 4 x₂ = 0
y₁ = 6 y₂ = -2
Midpoint (Center) = [tex]( \frac{4+0}{2}, \frac{6+ (-2)}{2}) [/tex]
= (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 = [tex]( \sqrt{(x_{2}-x_{1} )^{2} + (y_{2} -y_{1} ) ^{2}) [/tex]/2
Radius = [tex]( \sqrt{(0-4) ^{2}+(-2-6) ^{2} })/2 [/tex]
Radius = [tex]( \sqrt{(4) ^{2}+(-8) ^{2} })/2 [/tex]
Radius = [tex](1/2) \sqrt{(16)(5)} [/tex]
Radius = (1/2)(4) [tex] \sqrt{5} [/tex]
Radius = [tex]2 \sqrt{5} [/tex]
3) Equation:
Standard or Center-Radius Form:
(x - h)² + (y-h)² = r²
(x - 2)² + (y - 2)² = ([tex](2 \sqrt{5} ) ^{2} [/tex]
(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
Given: (4,6) and 0,-2)
1) Find the center (h, k) using midpoint formula.
Midpoint = [tex]( \frac{x_{1}+x_{2} }{2}, \frac{y_{1}+y_{2} }{2} )[/tex]
x₁ = 4 x₂ = 0
y₁ = 6 y₂ = -2
Midpoint (Center) = [tex]( \frac{4+0}{2}, \frac{6+ (-2)}{2}) [/tex]
= (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 = [tex]( \sqrt{(x_{2}-x_{1} )^{2} + (y_{2} -y_{1} ) ^{2}) [/tex]/2
Radius = [tex]( \sqrt{(0-4) ^{2}+(-2-6) ^{2} })/2 [/tex]
Radius = [tex]( \sqrt{(4) ^{2}+(-8) ^{2} })/2 [/tex]
Radius = [tex](1/2) \sqrt{(16)(5)} [/tex]
Radius = (1/2)(4) [tex] \sqrt{5} [/tex]
Radius = [tex]2 \sqrt{5} [/tex]
3) Equation:
Standard or Center-Radius Form:
(x - h)² + (y-h)² = r²
(x - 2)² + (y - 2)² = ([tex](2 \sqrt{5} ) ^{2} [/tex]
(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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