Question:
<p>An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points?</p>
<p style="display:inline"><span class="math-tex">\((1,2\sqrt{2})\)</span></p>
<p style="display:inline"><span class="math-tex">\((2,2\sqrt{2})\)</span></p>
<p style="display:inline"><span class="math-tex">\((\sqrt{2}, 2)\)</span></p>
<p style="display:inline"><span class="math-tex">\((2, \sqrt{2})\)</span></p>
Step-by-Step Solution
Key Concept: Identify the ellipse's orientation from its foci to correctly apply the relationship between the semi-major axis, semi-minor axis, and eccentricity.
<p>Let the equation of ellipse be<br />
<span class="math-tex">\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)</span> ...(i)<br />
Since, foci are at (0, 2) and (0, -2), major axis is along the Y-axis.<br />
So, be = 2 ... (ii)<br />
[where e is the eccentricity of ellipse]<br />
and 2a = length of minor axis = 4 [given]<br />
<span class="math-tex">\(\Rightarrow\)</span> a = 2 ... (iii)<br />
<span class="math-tex">\(\because \quad e^{2}=1-\frac{a^{2}}{b^{2}}\)</span><br />
<span class="math-tex">\(\therefore \quad\left(\frac{2}{b}\right)^{2}=1-\frac{4}{b^{2}}\)</span> <span class="math-tex">\(\left[\because e=\frac{2}{b}\right]\)</span><br />
<span class="math-tex">\(\Rightarrow \quad \frac{8}{b^{2}}=1 \Rightarrow b^{2}=8\)</span><br />
Thus, equation of required ellipse is <span class="math-tex">\(\frac{x^{2}}{4}+\frac{y^{2}}{8}=1\)</span><br />
Now, from the option the ellipse <span class="math-tex">\(\frac{x^{2}}{4}+\frac{y^{2}}{8}=1\)</span> = 1 passes through the point <span class="math-tex">\((\sqrt{2}, 2)\)</span></p>
Correct Answer: C