Answer:
The general equation is 4x² + 4y² - 8 x + 19 y - 140 = 0
Explanation:
* Lets revise the general form of the equation of the circle
- The general form : x² + y² + Dx + Ey + F = 0, where D, E, F are constants
- To find the values of D, E , F we must make three equations contains
D, E , F and solve them simultaneous
- We can do that if we have three points on the circle to substitute
x and y in the equation by them
* Now lets solve the problem
- There are three point on the circle (-5 , 0) , (0 , 4) , (2 , 4)
∵ The equation of the circle is x² + y² + Dx + Ey + F = 0
- Lets use the first point
∵ point (-5 , 0) lies on the circle
∴ (-5)² + (0)² + D(-5) + E(0) + F = 0 ⇒ simplify
∴ 25 - 5D + F = 0 ⇒ isolate F
∴ F = 5D - 25 ⇒ (1)
- Lets use the second point
∵ point (0 , 4) lies on the circle
∴ (0)² + (4)² + D(0) + E(4) + F = 0 ⇒ simplify
∴ 16 + 4E + F = 0 ⇒ substitute F form (1)
∴ 16 + 4E + 5D - 25 = 0
∴ 5D + 4E - 9 = 0 ⇒ add 9 to both sides
∴ 5D + 4E = 9 ⇒ (2)
- Lets use the third point
∵ point (2 , 4) lies on the circle
∴ (2)² + (4)² + D(2) + E(4) + F = 0 ⇒ simplify
∴ 4 + 16 + 2D + 4E + F = 0 ⇒ simplify
∴ 20 + 2D + 4E + F = 0 ⇒ substitute F form (1)
∴ 20 + 2D + 4E + 5D - 25 = 0 ⇒ simplify
∴ 7D + 4E - 5 = 0 ⇒ add 5 to both sides
∴ 7D + 4E = 5 ⇒ (3)
- Subtract (3) from (2) to eliminate E
∴ -2D = 4 ⇒ ÷ -2
∴ D = -2
- Substitute the value of D in (3) to find E
∴ 7(-2) + 4E = 5
∴ -14 + 4E = 5 ⇒ add 14 from both sides
∴ 4E = 19 ⇒ ÷ 4
∴ E = 19/4
- Substitute the value of D in (1) to find F
∴ F = 5(-2) - 25
∴ F = -10 - 25 = -35
∴ F = -35
* lets write the equation of the circle
∴ x² + y² - 2x + 19/4 y - 35 = 0 ⇒ × 4
∴ 4x² + 4y² - 8 x + 19 y - 140 = 0
* The general equation is 4x² + 4y² - 8 x + 19 y - 140 = 0