Solution
Step 1: General Equation of a Circle
The general equation of a circle is:
The centre of this circle is (−g,−f).
Step 2: Use the Condition for the Centre
The centre (−g,−f) lies on the line x+2y+3=0. Substituting the centre into the line equation:
−g−2f+3=0⟹g+2f=3— (Equation 1)
Step 3: Pass the Circle Through Given Points
The circle passes through (−1,1) and (2,1).
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For (−1,1):
(−1)2+(1)2+2g(−1)+2f(1)+c=0
1+1−2g+2f+c=0⟹−2g+2f+c=−2— (Equation 2)
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For (2,1):
(2)2+(1)2+2g(2)+2f(1)+c=0
4+1+4g+2f+c=0⟹4g+2f+c=−5— (Equation 3)
Step 4: Solve the Equations
Subtract (Equation 2) from (Equation 3) to eliminate f and c:
(4g+2f+c)−(−2g+2f+c)=−5−(−2)
Substitute g=−21 into (Equation 1):
−21+2f=3⟹2f=3.5=27⟹f=47
Substitute g and f into (Equation 2) to find c:
1+27+c=−2⟹c=−2−1−3.5=−6.5=−213
Step 5: Form the Final Equation
Substitute g=−21, f=47, and c=−213 into the general equation:
x2+y2+2(−21)x+2(47)y−213=0
Multiply the entire equation by 2 to clear the denominators:
Final Answer: The equation is 2x2+2y2−2x+7y−13=0.
