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Find the equation of the circle that passes through the points A(2, 0), B(2, 3),C (1, 3).

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Answer to a math question Find the equation of the circle that passes through the points A(2, 0), B(2, 3),C (1, 3).

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Darrell
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99 Answers
Solution:
1. Dado:
- Puntos A(2, 0), B(2, 3), C(1, 3)

2. La ecuación de una circunferencia en su forma general es:
x^2 + y^2 + Dx + Ey + F = 0

3. Sustituimos los puntos en la ecuación de la circunferencia para obtener un sistema de ecuaciones:
- Para A(2, 0):
2^2 + 0^2 + 2D + 0E + F = 0
4 + 2D + F = 0

- Para B(2, 3):
2^2 + 3^2 + 2D + 3E + F = 0
4 + 9 + 2D + 3E + F = 0
13 + 2D + 3E + F = 0

- Para C(1, 3):
1^2 + 3^2 + D + 3E + F = 0
1 + 9 + D + 3E + F = 0
10 + D + 3E + F = 0

4. Resolver el sistema de ecuaciones:
- Primer ecuación:
2D + F = -4

- Segunda ecuación:
13 + 2D + 3E + F = 0
2D + 3E + F = -13

- Tercer ecuación:
10 + D + 3E + F = 0
D + 3E + F = -10

5. Restar la primera ecuación de la segunda y la primera de la tercera:
- Segunda menos la primera:
(2D + 3E + F) - (2D + F) = -13 - (-4)
3E = -9
E = -3

- Tercera menos la primera:
(D + 3E + F) - (2D + F) = -10 - (-4)
-D + 3E = -6
-D + 3(-3) = -6
-D - 9 = -6
-D = 3
D = -3

6. Sustituir los valores de D y E en la primera ecuación:
2(-3) + F = -4
-6 + F = -4
F = 2

7. Con los valores D, E y F la ecuación de la circunferencia es:
x^2 + y^2 - 3x - 3y + 2 = 0

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