Find a positive value of (alpha) if the coordinate of the centre of a circle x(^2) + y(^2) – 2(alpha)x + 4y – (alpha) = 0 is ((alpha), -2) and the radius is 4 units.
-
A.
1 -
B.
2 -
C.
3 -
D.
4
Correct Answer: Option C
Explanation
(x^{2} + y^{2} – 2alpha x + 4y – alpha = 0)
r = 4 units; centre ((alpha), -2).
((x – x_{1})^{2} + (y – y_{1})^{2} = r^{2})
((x – alpha)^{2} + (y – (-2))^{2} = 4^{2})
(x^{2} – 2alpha x + alpha^{2} + y^{2} + 4y + 4 = 16)
(x^{2} + y^{2} – 2alpha x + 4y = 16 – alpha^{2} – 4)
(therefore 12 – alpha^{2} = alpha implies alpha^{2} + alpha – 12 = 0)
(alpha^{2} – 3alpha + 4alpha – 12 = 0)
(alpha(alpha – 3) + 4(alpha – 3) = 0)
((alpha + 4)(alpha – 3) = 0 implies +alpha = 3)