If the points (2,0) (0,1) (4,0) and (0,a)are concyclic then a=
Answers
Answer:
a = 1 or a = 8
Step-by-step explanation:
Find the equation of circle passes that passes through points(2,0) ,(0,1) and (4,0)
Equation of the circle is;
x^2 +y^2 +2gx +2fy +c =0
The equations for each of the points are;
(2,0):
(2)^2 + (0)^2 + 2g(2) + 2f(0) + c = 0
4 + 4g + c = 0 ………. Eq (1)
(0,1):
1 + 2f + c = 0 ………. Eq (2)
(4,0):
16 + 8g + c = 0 ………. Eq (3)
Now, Solving the equations:
Solving eq (1) & (3):
4 + 4g + c = 0
16 + 8g + c = 0
——————-
12 + 4g = 0
4g = - 12
g = -3
Solving eq (3):
16 + 8g + c = 0
16 + 8(-3) + c = 0
-8 + c = 0
c = 8
Solving eq(2):
1 + 2f + c = 0
1 + 2f + 8 = 0
2f = -9
f = -9/2
Therefore, the equation of the circle is
x^2 + y^2 + 2(-3)x + 2(-9/2)y + 8 = 0
x^2 + y^2 - 6x - 9y + 8 = 0
Now, (0,a) satisfies this equation;
Substituting the point (0,a) in the above equation;
(0)^2 + (a)^2 - 6(0) - 9(a) + 8 = 0
(a)^2 - 9a + 8 = 0
(a)^2 - 8a - a + 8 = 0
a(a - 8) - 1 (a - 8) = 0
(a - 1)(a - 8) = 0
(a - 1) = 0 or (a - 8) = 0
a = 1 or a = 8