question 7 of 10

the length of the transverse axis is 6, and the length of the red line segment is 15. how long is the blue line segment?

a. 6
b. 9
c. 15
d. 21

Question

question 7 of 10

the length of the transverse axis is 6, and the length of the red line segment is 15. how long is the blue line segment?

a. 6
b. 9
c. 15
d. 21

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Hyperbola Definition", "Hyperbola Axes" ], "new_concepts": [], "current_concepts": [ "Hyperbola Definition", "Hyperbola Axes", "Transverse Axis" ] } ## Identify the given values and hyperbola definition $$ \begin{aligned} &\text{Transverse axis length } (2a) = 6 \ &\text{Distance to the left focus } (d_1) = 15 \ &|d_1 - d_2| = 2a \end{aligned} $$ ## Calculate the length of the blue line segment $$ \begin{aligned} &|15 - d_2| = 6 \ &15 - d_2 = 6 \implies d_2 = 9 \ &15 - d_2 = -6 \implies d_2 = 21 \end{aligned} $$ ## Determine the geometrically valid solution $$ \begin{aligned} &\text{Since the point lies on the right branch, it is closer to the right focus:} \ &d_2 < d_1 \implies d_2 = 9 \end{aligned} $$

A. 6

B. 9

C. 15

D. 21

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Hyperbola Definition" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given values and hyperbola definition

Calculate the length of the blue line segment

Determine the geometrically valid solution

Answer: