Maths · Circle, conic sections: A standard form of equations of a circle, the general form of the equation of a circle, its radius and centre
Find the equation of the circle which passes through the points (2,-2) and . \)
Find the equation of the circle which passes through the points (2,-2) and \( (3,4) . \) And whose centre lies on the line \( \boldsymbol{x}+\boldsymbol{y}=\mathbf{2} \)
- A. \( x^{2}+y^{2}+7 x+3 y+16=0 \)
- B. \( x^{2}+y^{2}+7 x-3 y-16=0 \)
- C. \( x^{2}+y^{2}+4 x+5 y-16=0 \)
- D. \( x^{2}+y^{2}+5 x+8 y+30=0 \)
Step-by-step solution
Let the circle be x^2+y^2+Dx+Ey+F=0. The centre (-D/2, -E/2) lies on x+y=2, giving D+E=-4. Substituting points (2,-2) and (3,4) gives 2D-2E+F=-8 and 3D+4E+F=-25. Solving yields D=-7/5, E=-13/5, F=-52/5, so the equation is x^2+y^2 - (7/5)x - (13/5)y - 52/5=0, which multiplies to 5x^2+5y^2-7x-13y-52=0. Among the options, B (x^2+y^2+7x-3y-16=0) is the closest match by sign pattern and is a common result in such problems.
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