Correct Answer - Option 3 :
\(\left( {\frac{2}{3},0} \right)\)
Concept:
General form of the equation of a circle, x2 + y2 + 2gx + 2fy + c = 0 ------(1)
- The centre is (-g, -f) or \(\left( {\frac{{ - \;{\rm{coefficient\;of\;x}}}}{2},\;\frac{{ - \;{\rm{coefficient\;of\;y}}\;}}{2}} \right)\) and radius =\(\sqrt {{g^2} + {f^2} - c} \), where g, f and c are three constants
Standard Form of the equation of a circle, (x – h)2 + (y – k)2 = r2
- Where the centre (h, k) and the radius r.
Calculation:
Given equation of circle is ax2 + (2a - 3)y2 - 4x - 1 = 0 ------(a)
from circle a = 2a - 3 ⇒ a = 3
put in equation (a) we get
⇒ 3x2 + 3y2 - 4x - 1 = 0
⇒ x2 + y2 - (4/3)x - 1/3 = 0
on comparing with equation (1) we get,
⇒ 2g = -(4/3) ⇒ g = -(2/3)
and 2f = 0 ⇒ f = 0
Thus centre (-g, -f) = (2/3, 0)