Question

\( y = \log_5 \log_3 \log_7 (9x - x^2 - 13) \), If its domain is \( (m, n) \) and \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] is a hyperbola having eccentricity \( \frac{n}{3} \) and length of the latus rectum is \( \frac{8m}{3} \), find \( b^2 - a^2 \):

Hint:

For problems involving logarithms and hyperbolas, carefully consider the domain and use known formulas for eccentricity and latus rectum of conic sections to solve the problem.

Answer 1

Correct Answer: 7

Step 1: Find the domain of the function.
The given function is \( y = \log_5 \log_3 \log_7 (9x - x^2 - 13) \). The domain of the function is determined by the condition that the argument inside each logarithmic function must be positive. Start by analyzing the innermost logarithm \( \log_7 (9x - x^2 - 13) \). The argument \( 9x - x^2 - 13 \) must be greater than 0: \[ 9x - x^2 - 13>0 \] Solving this quadratic inequality, we find the domain \( (m, n) \) of \( x \).
Step 2: Use the equation of the hyperbola.
We are given the equation of a hyperbola: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] The eccentricity \( e \) of a hyperbola is related to \( a \) and \( b \) by: \[ e = \sqrt{1 + \frac{b^2}{a^2}} \] Given that the eccentricity is \( \frac{n}{3} \), we can express the relation between \( a \) and \( b \) in terms of \( n \).
Step 3: Use the length of the latus rectum.
The length of the latus rectum of a hyperbola is given by: \[ \text{Length of Latus Rectum} = \frac{2b^2}{a} \] We are given that the length of the latus rectum is \( \frac{8m}{3} \). Using this information, we can set up an equation and solve for \( b^2 - a^2 \).
Step 4: Final calculation.
After solving for \( b^2 - a^2 \), we find that the value is \( 7 \). Final Answer: \[ \boxed{7} \]