Explanation
Elimination Using Options (Fastest Method)
A circle passing through specific points must satisfy the equation of the circle when those coordinates are substituted into it. Let's test the given points:
Step 1: Eliminate Option (b)
Option (b) contains a −y2 term (x2−y2+7x−5y+12=0). In standard circle equations, the coefficients of x2 and y2 must be equal and have the same sign. Thus, option (b) represents a hyperbola, not a circle.
Step 2: Test the point (−1,1) in Option (a)
(−1)2+(1)2−7(−1)+5(1)−14=0
1+1+7+5−14=0
14−14=0(Satisfied!)
Step 3: Test the point (2,3) in Option (a)
(2)2+(3)2−7(2)+5(3)−14=0
4+9−14+15−14=0
28−28=0(Satisfied!)
Since option (a) satisfies both given points, it is the correct circle equation.
