Write the equation of the family of all concentric circles whose common center is the point (-3,

step 1 here we have the center already but we are missing the radius here so in the to find the radius

Write the equation of the family of all concentric circles...

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Key Concepts

Standard Equation of a Circle

The standard equation of a circle in the x-y plane is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is its radius. This form is essential for defining and analyzing the properties of a circle in coordinate geometry.

Concentric Circles

Concentric circles are circles that share the same center point but have different radii. Although they have a common center, the varying radius gives each circle a unique size, which is a key concept when considering a family of circles in the same geometric space.

Parameterization in Circle Equations

When writing a family of curves, introducing a parameter allows for the representation of multiple figures with a single equation. In the context of concentric circles, the radius (often represented as a parameter such as r or k) serves as the variable that distinguishes one circle from another.

Graphical Representation of Circle Families

Drawing different members of a family of circles involves selecting specific values for the parameter, generally the radius, to visualize how the circles vary in size while maintaining the same center. This aids in understanding the geometric relationships and scaling between the circles.

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