To determine the correct general form of the equation of a circle with a given center and radius, let’s begin by recalling the standard form of a circle's equation.
A circle with center [tex]\((a, b)\)[/tex] and radius [tex]\(m\)[/tex] is represented in its standard form as:
[tex]\[ (x - a)^2 + (y - b)^2 = m^2 \][/tex]
To transform this standard form into a general form, we need to expand and simplify the equation.
Starting with:
[tex]\[ (x - a)^2 + (y - b)^2 = m^2 \][/tex]
Expand the squared terms:
[tex]\[ (x^2 - 2ax + a^2) + (y^2 - 2by + b^2) = m^2 \][/tex]
Combine and rearrange all terms on one side of the equation to set it to zero:
[tex]\[ x^2 - 2ax + a^2 + y^2 - 2by + b^2 - m^2 = 0 \][/tex]
Now, group like terms together:
[tex]\[ x^2 + y^2 - 2ax - 2by + (a^2 + b^2 - m^2) = 0 \][/tex]
Examining the provided options:
A. [tex]\( x^2 + y^2 - 2ax - 2by + (a^2 + b^2 - m^2) = 0 \)[/tex]
B. [tex]\( x^2 + y^2 + 2ax + 2by + (a^2 + b^2 - m^2) = 0 \)[/tex]
C. [tex]\( x^2 + y^2 - 2ax - 2by + (a + b - m^2) = 0 \)[/tex]
D. [tex]\( x^2 + y^2 + 2ax + 2by + a^2 + b^2 = -m^2 \)[/tex]
The correct general form that matches our derived equation is:
A. [tex]\( x^2 + y^2 - 2ax - 2by + (a^2 + b^2 - m^2) = 0 \)[/tex]
Thus, the correct answer is:
A. [tex]\(x^2 + y^2 - 2ax - 2by + (a^2 + b^2 - m^2) = 0 \)[/tex]
