JEE Mains · Maths · STD 11 - 10.2 parabola,ellipse,hyperbola
Let the domain of the function \( f(x)=\log_{3}\log_{5}\log_{7}(9x-x^{2}-13) \) be the interval (m, n). Let the hyperbola \( \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) have eccentricity \( \frac{n}{3} \) and the length of the latus rectum \( \frac{8m}{3} \). Then \( b^{2}-a^{2} \) is equal to:
- A 5
- B 11
- C 9
- D 7
Answer & Solution
Correct Answer
(D) 7
Step-by-step Solution
Detailed explanation
\(\log _5\left(\log _7\left(9 x-x^2-13\right)\right)>0\) \(\Rightarrow \quad 9 x-x^2-13>7\) \(x^2-9 x+20 < 0 \Rightarrow 4< x< 5\) \(m =4, n =5\) \(\Rightarrow \quad e =\sqrt{1+\frac{ b ^2}{ a ^2}}=\frac{5}{3} \Rightarrow \frac{b^2}{ a ^2}=\frac{25}{9}-1=\frac{16}{9}\)…
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