To write down the equation of a circle in the form [tex]\( ax^2 + bx + cy^2 + dy + e = 0 \)[/tex], we start with the standard form of the equation of a circle and proceed to expand and rearrange it as required.
The standard form of a circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where:
- [tex]\((h, k)\)[/tex] are the coordinates of the center of the circle,
- [tex]\(r\)[/tex] is the radius of the circle.
Let's expand and rearrange this equation step-by-step:
1. Expand the squared terms:
[tex]\[ (x - h)^2 = x^2 - 2hx + h^2 \][/tex]
[tex]\[ (y - k)^2 = y^2 - 2ky + k^2 \][/tex]
2. Substitute these expanded forms back into the standard equation:
[tex]\[ x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2 \][/tex]
3. Combine like terms and move all terms to one side of the equation to set it equal to zero:
[tex]\[ x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0 \][/tex]
4. Identify and write down the coefficients for the general form:
[tex]\[ x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0 \][/tex]
This equation is now in the desired form [tex]\( ax^2 + bx + cy^2 + dy + e = 0 \)[/tex], where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2h \)[/tex]
- [tex]\( c = 1 \)[/tex]
- [tex]\( d = -2k \)[/tex]
- [tex]\( e = h^2 + k^2 - r^2 \)[/tex]
Thus, the general form of the circle's equation is:
[tex]\[ x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0 \][/tex]
Let's work through a specific example. Suppose the center of the circle is [tex]\((3, -4)\)[/tex] and the radius is [tex]\(5\)[/tex].
Step-by-step for this example:
1. Start with the standard form:
[tex]\[ (x - 3)^2 + (y + 4)^2 = 5^2 \][/tex]
2. Expand the squared terms:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
[tex]\[ (y + 4)^2 = y^2 + 8y + 16 \][/tex]
3. Combine the expanded terms:
[tex]\[ x^2 - 6x + 9 + y^2 + 8y + 16 = 25 \][/tex]
4. Move all terms to one side:
[tex]\[ x^2 + y^2 - 6x + 8y + 9 + 16 - 25 = 0 \][/tex]
5. Combine like terms:
[tex]\[ x^2 + y^2 - 6x + 8y = 0 \][/tex]
So, the equation of the circle in the form [tex]\( ax^2 + bx + cy^2 + dy + e = 0 \)[/tex] for this specific example is:
[tex]\[ x^2 + y^2 - 6x + 8y = 0 \][/tex]
