Exercises: Derive the Equation of a Circle and Find Center and Radius

Show your work for each problem. Express equations in standard form unless otherwise stated.

Grade 10·22 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-gpe-a-1

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Warm-Up: Review What You Know

These problems review skills you have already learned.

What is the distance between the points $(1, 2)$ and $(4, 6)$ ?

$\sqrt{7}$

A circle is defined as the set of all points in a plane that are:

Inside a fixed boundary

A fixed distance from a center point

Connected by a curved line

Equidistant from two fixed points

Complete the square: $x^2 - 10x + $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $= (x - 5)^2$ . What number fills the blank?

Write the equation of the circle with center $(3, -2)$ and radius $7$ in standard form.

What is the center of the circle given by $(x + 4)^2 + (y - 1)^2 = 36$ ?

$(4, 1)$

$(-4, -1)$

$(-4, 1)$

$(4, -1)$

What is the radius of the circle given by $(x - 5)^2 + (y + 3)^2 = 64$ ?

Find the center and radius of the circle $x^2 + y^2 - 6x + 8y - 11 = 0$ . Enter the center as $(h, k)$ and the radius as $r$ .

Convert $x^2 + y^2 + 10x - 4y + 20 = 0$ to standard form and identify the radius.

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