Mathematics
Grade 10
15 min
Write equations of circles in standard form using properties
Write equations of circles in standard form using properties
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the center and radius of a circle from its standard form equation.
Write the standard form equation of a circle given its center and radius.
Write the standard form equation of a circle given its center and a point on the circle.
Write the standard form equation of a circle given the endpoints of a diameter.
Derive the equation of a circle using the Distance Formula.
Determine if a given point lies on, inside, or outside a circle using its equation.
Ever wonder how your phone's GPS pinpoints your location or how a seismograph locates an earthquake's epicenter? 🗺️ It all starts with the simple, powerful equation of a circle!
In this tutorial, you will learn how to translate the geometric properties of a circle—its center and radius—into...
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Key Concepts & Vocabulary
TermDefinitionExample
CircleThe set of all points in a plane that are a fixed distance (the radius) from a fixed point (the center).All the points that are exactly 5 units away from the point (2, 3) form a circle.
CenterThe fixed point, denoted as (h, k), from which all points on the circle are equidistant.For a circle with the equation (x - 4)^2 + (y + 1)^2 = 25, the center is at (4, -1).
RadiusThe fixed distance, denoted as r, from the center to any point on the circle. It is always a positive value.For a circle with the equation (x - 4)^2 + (y + 1)^2 = 25, the radius is √25, which is 5.
DiameterA line segment that passes through the center of a circle and has its endpoints on the circle. Its length is twice the radius (d = 2r).If a circle has a radius of 6, its diameter is 12.
Standard...
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Core Formulas
Standard Form of the Equation of a Circle
(x - h)^2 + (y - k)^2 = r^2
This is the fundamental formula for a circle. (h, k) represents the coordinates of the center, and r represents the length of the radius. Notice the minus signs with h and k; this means you take the opposite sign of the numbers in the parentheses to find the center's coordinates.
Distance Formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Use this formula to find the radius (r) when you are given the center (h, k) and another point on the circle (x, y). The distance 'd' will be the radius 'r'.
Midpoint Formula
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
Use this formula to find the center of a circle (h, k) when you are given the two endpoints of a diameter, (x₁...
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Challenging
A circle has a diameter with endpoints at (1, √2) and (5, √2). It is then translated 2 units down. What is the equation of the translated circle?
A.(x - 3)² + (y - √2 + 2)² = 4
B.(x - 3)² + (y + 2)² = 4
C.(x - 3)² + (y - (√2 - 2))² = 4
D.(x - 3)² + (y - √2)² = 16
Challenging
A circle has its center at (h, k) and is tangent to the line y = -1. If the circle also passes through the point (h, 5), what is its equation?
A.(x - h)² + (y - k)² = 9
B.(x - h)² + (y - 2)² = 9
C.(x - h)² + (y - k)² = 36
D.(x - h)² + (y - 2)² = 36
Challenging
The center of a circle is in the first quadrant. The circle is tangent to the x-axis, the y-axis, and the line x = 8. What is the equation of the circle?
A.(x - 4)² + (y - 4)² = 16
B.(x - 8)² + (y - 8)² = 64
C.(x - 4)² + (y - 4)² = 4
D.(x - 8)² + (y - 4)² = 16
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