Question

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

Explanation:

Step1: Recall hyperbola property

For a hyperbola, the difference of the distances from any point on the hyperbola to the two foci is equal to the length of the transverse axis. Let the length of the blue - line segment be $x$ and the length of the red - line segment be $y = 15$, and the length of the transverse axis be $2a=6$.

Step2: Set up the equation

The absolute value of the difference between the distances from a point on the hyperbola to the two foci is $|x - y|=2a$. Substituting $y = 15$ and $2a = 6$ into the equation, we get $|x - 15|=6$.

Step3: Solve the equation

Case 1: $x-15 = 6$, then $x=6 + 15=21$. Case 2: $x - 15=-6$, then $x=15 - 6 = 9$. Since the length of a line - segment is non - negative, and considering the geometric situation of the hyperbola, we have two possible values for $x$. But in the context of the hyperbola's focus - point distance relationship, if we assume the correct orientation and setup, we get $x=21$.

Answer:

B. 21