Question: The general form of the family of curves for circles is given by:
Options:
(x - h)^2 + (y - k)^2 = r^2
x^2 + y^2 = r^2
x^2 + y^2 + Dx + Ey + F = 0
y = mx + b
Correct Answer: x^2 + y^2 + Dx + Ey + F = 0
Solution:
The equation x^2 + y^2 + Dx + Ey + F = 0 represents a family of circles.
The general form of the family of curves for circles is given by:
Practice Questions
Q1
The general form of the family of curves for circles is given by:
(x - h)^2 + (y - k)^2 = r^2
x^2 + y^2 = r^2
x^2 + y^2 + Dx + Ey + F = 0
y = mx + b
Questions & Step-by-Step Solutions
The general form of the family of curves for circles is given by:
Step 1: Understand that a circle is defined by its center and radius.
Step 2: The general equation of a circle can be transformed into a standard form.
Step 3: The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
Step 4: Expand the standard form to get x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2.
Step 5: Rearrange the equation to the form x^2 + y^2 + Dx + Ey + F = 0, where D = -2h, E = -2k, and F = h^2 + k^2 - r^2.
Step 6: Recognize that D, E, and F can vary, leading to different circles, hence the term 'family of curves'.
Circle Equation – The general equation of a circle in a Cartesian coordinate system, which can be manipulated to represent different circles based on the values of D, E, and F.
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