What is the radius of a circle given the equation $$x^2 + y^2 + Dx + Ey + F = 0$$ ?

The radius can be found by rewriting the equation in standard form and identifying r from $$(x-h)^2 + (y-k)^2 = r^2$$ .

How do you determine the radius from the equation $$x^2 + y^2 - 10x + 8y - 23 = 0$$ ?

Rewrite the equation in standard form to find the radius, which is $$8$$ .

What is the significance of the values h and k in the circle's equation?

The values h and k represent the x and y coordinates of the center of the circle.

How do you convert the equation $$x^2 + y^2 - 11 - 10x = 0$$ to standard form?

Complete the square for x and y to rewrite it as $$(x-5)^2 + y^2 = 36$$ .

What is the radius of the circle represented by the equation $$(x+1)^2 + (y+1)^2 = 9$$ ?

The radius is $$3$$ , since $$r^2 = 9$$ .

What does the term 'completing the square' help to achieve in circle equations?

It helps to rewrite the equation in standard form, making it easier to identify the center and radius.

What is the general form of a circle's equation?

The general form is $$x^2 + y^2 + Dx + Ey + F = 0$$ .

How can you identify the radius from the standard form of a circle's equation?

The radius is the square root of the constant on the right side of the equation, $$r = \text{sqrt}(r^2)$$ .

What is the relationship between the coefficients in the general form and the center of the circle?

The coefficients D and E in the general form relate to the center's coordinates through the equations h = -D/2 and k = -E/2.

10.7 Equations of Circles with Completing the Square FLASHCARD Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

What is the standard form of a circle's equation?

Back

2.

FLASHCARD QUESTION

Front

How do you find the center of a circle from its equation?

Back

3.

FLASHCARD QUESTION

Front

Back

4.

FLASHCARD QUESTION

Front

What does completing the square involve?

Back

Completing the square involves rearranging a quadratic equation to form a perfect square trinomial.

5.

FLASHCARD QUESTION

Front

Back

6.

FLASHCARD QUESTION

Front

Back

7.

FLASHCARD QUESTION

Front

What is the equation of a circle in standard form?

Back

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