Question:
<p>If <span class="math-tex">\(S\)</span> and <span class="math-tex">\(S^{\prime}\)</span> are the foci of the ellipse <span class="math-tex">\(\frac{x^{2}}{18}+\frac{y^{2}}{9}\)</span> <span class="math-tex">\(=1\)</span> and P be a point on the ellipse, then <span class="math-tex">\(\min \left({SP} \cdot {S}^{\prime} {P}\right)+\max \left({SP} \cdot {S}^{\prime} {P}\right)\)</span> is equal to:</p>
<p style="display:inline"><span class="math-tex">\(3(1+\sqrt{2})\)</span></p>
<p style="display:inline"><span class="math-tex">\(3(6+\sqrt{2})\)</span></p>
<p style="display:inline">27</p>
<p style="display:inline">9</p>
Step-by-Step Solution
Key Concept: For an ellipse with foci S and S', use the property SP + S'P = 2a for any point P on the ellipse. The product SP·S'P is extremized when P is at the endpoints of major or minor axes, requiring calculation using c² = a² - b² and the focal radius formula r = a ± ex.
<p><span class="math-tex">\(\frac{x^{2}}{18}+\frac{y^{2}}{9}=1\)</span><br />
<span class="math-tex">\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)</span><br />
On comparison, we get<br />
<span class="math-tex">\(a^{2}=18, b^{2}=9 \Rightarrow a \gt b\)</span><br />
<img src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1757573764-2bw5zc.jpg" style="height:138px; width:300px" /><br />
<span class="math-tex">\({PS}+{PS}^{\prime}=2 \times 3 \sqrt{2}\)</span><br />
<span class="math-tex">\(b^{2}=a^{2}\left(1-e^{2}\right)=18\left(1-e^{2}\right)\)</span><br />
<span class="math-tex">\(e=\frac{1}{\sqrt{2}}\)</span><br />
<span class="math-tex">\(\text { Directrix }\)</span> <span class="math-tex">\(x =\frac{3 \sqrt{2}}{\frac{1}{\sqrt{2}}}=6\)</span><br />
<span class="math-tex">\({SP} =a(1-e \cos \theta)\)</span><br />
<span class="math-tex">\({S}^{\prime} {P} =a(1+e \cos \theta)\)</span><br />
<span class="math-tex">\({SP} \cdot {~S}^{\prime} {P} =a^{2}(1-e \cos \theta)(1+e \cos \theta)\)</span><br />
<span class="math-tex">\(=a^{2}\left(1-e^{2} \cos { }^{2} \theta\right)\)</span><br />
<span class="math-tex">\(=18\left(1-\frac{\cos ^{2} \theta}{2}\right)\)</span><br />
<span class="math-tex">\(\left({SP} \cdot {~S}^{\prime} {P}\right)_{\max } =18 \text { if } \cos \theta=0\)</span><br />
<span class="math-tex">\(\left({SP} \cdot {~S}^{\prime} {P}\right)_{\min } =9 \text { if } \cos \theta=1\)</span><br />
<span class="math-tex">\({Sum} =18+9=27\)</span></p>
Correct Answer: C