Coordinate Geometry: Circle Equation

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Consider a circle with radius=10 units and center C(0,0). Point A(6, 8) and Point Q(-8, -6) exist on the circle.

Draw the radius to Point A.
Draw the vertical line segment \overline{AB} to the x-axis.
Draw the horizontal line segment \overline{BC} to the y-axis.

You have just created right triangle \triangle ABC. Repeat these steps to create right triangle \triangle QPC.

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Use the Pythagorean Theorem to relate the three side lengths of the right triangles.

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The coordinates from A(6,8) satisfy the Pythagorean Theorem:

The coordinates from Q(-8,-6), although they are negative, also satisfy the Pythagorean Theorem:

With this information in mind, what Standard Equation can you write to represent the equation of a circle with center (0,0) and radius r?

Any point with coordinates that satisfy this Standard Equation exists ON the circle.  However, it is possible to determine if a point exists INSIDE the circle or OUTSIDE the circle as well!

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Given the equation:
conjecture if the points L(4,-2), M(0, -8), and N(8, -7) exist INSIDE the circle, ON the circle, or OUTSIDE the circle.

(You may graph within the Show Your Work space if you would find it helpful!)

Draggable item		Corresponding Item
N(8, -7)		INSIDE the circle
L(4, -2)		ON the circle
M(0, -8)		OUTSIDE the circle

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What is the radius of the circle in each equation?

Draggable item		Corresponding Item

The circle below is called the Unit Circle, because it has radius=1 unit and center (0,0).
In this investigation, you will learn how to transform the Unit Circle, both graphically and symbolically.

Unit Circle Equation:
Since ALL circles are similar, EVERY circle is a transformation of the Unit Circle.  Both equations below represent a "scaled up" version of the Unit Circle (scale factor of 5).  Notice the Standard Form is set equal to r^2.  The Transformation Form is set equal to 1, and the scale factor appears in the denominator beneath x and y.

Standard Form:

Transformation Form:

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Draw the circle that is represented by this equation in Transformation Form:

A circle can also be transformed by translation.  Both equations below represent a "scaled up" version of the Unit Circle (scale factor of 4) that has also been translated 3 units left and 5 units up.

Standard Form:

Transformation Form:

IMPORTANT:  The operations exist INSIDE the (      ), so we perform the INVERSE transformation of what may seem to make sense!  The new center is NOT (+3, -5) as you may suspect!   The new center is actually (-3, +5).

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Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.

Equation:

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Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.

Equation:

But what happens when the horizontal scale change and the vertical scale change have DIFFERENT scale factors?  In this situation, a new shape is formed.  The circle is transformed into an ellipse!

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Match the equation of each ELLIPSE to its graph.

Draggable item		Corresponding Item

Coordinate Geometry: Circle Equation

Use the Pythagorean Theorem to relate the three side lengths of the right triangles.

The coordinates from A(6,8) satisfy the Pythagorean Theorem:The coordinates from Q(-8,-6), although they are negative, also satisfy the Pythagorean Theorem:With this information in mind, what Standard Equation can you write to represent the equation of a circle with center (0,0) and radius r?

Given the equation:conjecture if the points L(4,-2), M(0, -8), and N(8, -7) exist INSIDE the circle, ON the circle, or OUTSIDE the circle.(You may graph within the Show Your Work space if you would find it helpful!)

What is the radius of the circle in each equation?

Draw the circle that is represented by this equation in Transformation Form:

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.Equation:

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.Equation:

Match the equation of each ELLIPSE to its graph.

Use the Pythagorean Theorem to relate the three side lengths of the right triangles.

The coordinates from A(6,8) satisfy the Pythagorean Theorem:The coordinates from Q(-8,-6), although they are negative, also satisfy the Pythagorean Theorem:With this information in mind, what Standard Equation can you write to represent the equation of a circle with center (0,0) and radius r?

Given the equation:conjecture if the points L(4,-2), M(0, -8), and N(8, -7) exist INSIDE the circle, ON the circle, or OUTSIDE the circle.(You may graph within the Show Your Work space if you would find it helpful!)

What is the radius of the circle in each equation?

Draw the circle that is represented by this equation in Transformation Form:

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.Equation:

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.Equation:

Match the equation of each ELLIPSE to its graph.

The coordinates from A(6,8) satisfy the Pythagorean Theorem:

The coordinates from Q(-8,-6), although they are negative, also satisfy the Pythagorean Theorem:

With this information in mind, what Standard Equation can you write to represent the equation of a circle with center (0,0) and radius r?

Given the equation:
conjecture if the points L(4,-2), M(0, -8), and N(8, -7) exist INSIDE the circle, ON the circle, or OUTSIDE the circle.

(You may graph within the Show Your Work space if you would find it helpful!)

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.

Equation:

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.

Equation:

The coordinates from A(6,8) satisfy the Pythagorean Theorem:

The coordinates from Q(-8,-6), although they are negative, also satisfy the Pythagorean Theorem:

With this information in mind, what Standard Equation can you write to represent the equation of a circle with center (0,0) and radius r?

Given the equation:
conjecture if the points L(4,-2), M(0, -8), and N(8, -7) exist INSIDE the circle, ON the circle, or OUTSIDE the circle.

(You may graph within the Show Your Work space if you would find it helpful!)

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.

Equation:

Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.

Equation: