Question:
<p>Let one focus of the hyperbola. <span class="math-tex">\({H}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\)</span> <span class="math-tex">\(=1\)</span> be at <span class="math-tex">\((\sqrt{10}, 0)\)</span> and the corresponding directrix be <span class="math-tex">\(x=\frac{9}{\sqrt{10}}\)</span>. If <span class="math-tex">\(e\)</span> and <span class="math-tex">\(l\)</span> respectively are the eccentricity and the length of the latus rectum of H , then <span class="math-tex">\(9\left(e^{2}+l\right)\)</span> is equal to:</p>
<p style="display:inline">14</p>
<p style="display:inline">15</p>
<p style="display:inline">12</p>
<p style="display:inline">16</p>
Step-by-Step Solution
Key Concept: Solve for the parameters 'a' and 'e' using the focus and directrix definitions, then apply the hyperbola's fundamental relation to find the latus rectum.
<p><span class="math-tex">\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} =1\)</span><br />
<span class="math-tex">\(\text { focus } =(\sqrt{10}, 0)\)</span><br />
<span class="math-tex">\(\text { Transverse axis } =2 a\)</span><br />
<span class="math-tex">\(\text { Minor axis } =2 b\)</span><br />
<span class="math-tex">\(\text { Eccentricity } =e\)</span><br />
<span class="math-tex">\(\text { Directrix } \Rightarrow x =\frac{a}{e}\)</span><br />
<span class="math-tex">\(a e =\sqrt{10} \text { and } \frac{a}{e}=\frac{9}{\sqrt{10}}\)</span><br />
<span class="math-tex">\(\Rightarrow a^{2} =9 \text { and } e=\frac{\sqrt{10}}{3}\)</span><br />
<span class="math-tex">\(\text { Now, } (a e)^{2} =a^{2}+b^{2}\)</span><br />
<span class="math-tex">\(10 =9+b^{2}\)</span><br />
<span class="math-tex">\(\therefore b =1\)</span><br />
<span class="math-tex">\(\text { latus rectum }(l) =\frac{2 b^{2}}{a}=\frac{2 \times 1}{3}\)</span><br />
<span class="math-tex">\(\Rightarrow 9\left(e^{2}+1\right)\)</span><br />
<span class="math-tex">\(=9\left(\frac{10}{9}+\frac{2}{3}\right)\)</span><br />
<span class="math-tex">\(=10+6\)</span><br />
<span class="math-tex">\(=16\)</span></p>
Correct Answer: D