Question:
<p>The length of transverse axis of the hyperbola 3x<sup>2</sup> - 4y<sup>2</sup> = 32 is</p>
<p style="display:inline"><span class="math-tex">\(\frac{3}{32}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{8 \sqrt{2}}{\sqrt{3}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{16 \sqrt{2}}{\sqrt{3}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{64}{3}\)</span></p>
Step-by-Step Solution
Key Concept: Convert the hyperbola equation to standard form x^2/a^2 - y^2/b^2 = 1 to determine the semi-transverse axis length a and calculate the total transverse axis as 2a.
<p>The given equation can be written as<br />
<span class="math-tex">$\frac{x^{2}}{\frac{32}{3}}-\frac{y^{2}}{8}=1$</span> <span class="math-tex">$\Rightarrow \frac{x^{2}}{\left(\frac{4 \sqrt{2}}{\sqrt{3}}\right)^{2}}-\frac{y^{2}}{(2 \sqrt{2})^{2}}=1$</span><br />
Here, <span class="math-tex">$a^{2}=\left(\frac{4 \sqrt{2}}{\sqrt{3}}\right)^{2} \Rightarrow a=\frac{4 \sqrt{2}}{\sqrt{3}}$</span><br />
The length of transverse axis = 2a <span class="math-tex">$=2 \times \frac{4 \sqrt{2}}{\sqrt{3}}$</span> <span class="math-tex">$=\frac{8 \sqrt{2}}{\sqrt{3}}$</span></p>
Correct Answer: B