question 9 of 10

which of the following circles lie completely within the third quadrant?

check all that apply.

a. $(x + 3)^2 + (y + 9)^2 = 82$
b. $(x + 12)^2 + (y + 9)^2 = 9$
c. $(x + 7)^2 + (y + 7)^2 = 4$
d. $(x + 5)^2 + (y + 0)^2 = 7$

Question

question 9 of 10

which of the following circles lie completely within the third quadrant?

check all that apply.

a. $(x + 3)^2 + (y + 9)^2 = 82$
b. $(x + 12)^2 + (y + 9)^2 = 9$
c. $(x + 7)^2 + (y + 7)^2 = 4$
d. $(x + 5)^2 + (y + 0)^2 = 7$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Circle Definition", "Circle Radius" ], "new_concepts": [ "Circle Quadrant Location" ], "current_concepts": [ "Circle Definition", "Circle Radius", "Circle Quadrant Location" ] } ## Formulate the condition for a circle to lie completely in the third quadrant $$ \begin{aligned} &\text{Circle equation: } (x-h)^2 + (y-k)^2 = r^2 \implies \text{Center: } (h, k), \text{ Radius: } r = \sqrt{r^2}\ &\text{Third quadrant condition: } x < 0 \text{ and } y < 0\ &\text{For the entire circle to lie in the third quadrant: } h + r < 0 \text{ and } k + r < 0 \end{aligned} $$ ## Evaluate each option $$ \begin{aligned} &\text{Option A: } (x+3)^2 + (y+9)^2 = 82 \implies (h, k) = (-3, -9), r = \sqrt{82} \approx 9.06\ &h + r = -3 + 9.06 = 6.06 ot< 0 \quad (\text{Fails})\ \ &\text{Option B: } (x+12)^2 + (y+9)^2 = 9 \implies (h, k) = (-12, -9), r = 3\ &h + r = -12 + 3 = -9 < 0 \quad \text{and} \quad k + r = -9 + 3 = -6 < 0 \quad (\text{Passes})\ \ &\text{Option C: } (x+7)^2 + (y+7)^2 = 4 \implies (h, k) = (-7, -7), r = 2\ &h + r = -7 + 2 = -5 < 0 \quad \text{and} \quad k + r = -7 + 2 = -5 < 0 \quad (\text{Passes})\ \ &\text{Option D: } (x+5)^2 + (y+0)^2 = 7 \implies (h, k) = (-5, 0), r = \sqrt{7} \approx 2.65\ &k + r = 0 + 2.65 = 2.65 ot< 0 \quad (\text{Fails}) \end{aligned} $$ ## Identify the correct choices $$ \text{Correct options: B and C} $$

A. $(x + 3)^2 + (y + 9)^2 = 82$

B. $(x + 12)^2 + (y + 9)^2 = 9$

C. $(x + 7)^2 + (y + 7)^2 = 4$

D. $(x + 5)^2 + (y + 0)^2 = 7$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Circle Radius" ] }

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QUESTION IMAGE

Question

Explanation:

Formulate the condition for a circle to lie completely in the third quadrant

Evaluate each option

Identify the correct choices

Answer: