Question:
<p>Let the length of the latus rectum of an ellipse with its major axis along X-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?</p>
<p style="display:inline"><span class="math-tex">\((4 \sqrt{3}, 2 \sqrt{3})\)</span></p>
<p style="display:inline"><span class="math-tex">\((4 \sqrt{2}, 2 \sqrt{2})\)</span></p>
<p style="display:inline"><span class="math-tex">\((4 \sqrt{3}, 2 \sqrt{2})\)</span></p>
<p style="display:inline"><span class="math-tex">\((4 \sqrt{2}, 2 \sqrt{3})\)</span></p>
Step-by-Step Solution
Key Concept: Determine the ellipse's equation by translating the given geometric conditions of latus rectum length and focal distance into algebraic equations using the semi-axes $a, b$ and the eccentricity relationship $b^2 = a^2(1 - e^2)$.
<p>Let the equation of ellipse be <span class="math-tex">$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$</span><br />
Then, according the problem, we have<br />
<span class="math-tex">$\frac{2 b^{2}}{a}=8$</span> and 2ae = 2b [Length of latusrectum = <span class="math-tex">$\frac{2 b^{2}}{a}$</span> and length of minor axis = 2b]<br />
<span class="math-tex">$\Rightarrow \quad b\left(\frac{b}{a}\right)=4 \text { and } \frac{b}{a}=e$</span><br />
<span class="math-tex">$\Rightarrow$</span> b(e) = 4<br />
<span class="math-tex">$\Rightarrow \quad b=4 . \frac{1}{e}$</span> ...(i)<br />
Also, we know that b<sup>2</sup> = a<sup>2</sup>(1 - e<sup>2</sup>)<br />
<span class="math-tex">$\Rightarrow \quad \frac{b^{2}}{a^{2}}=1-e^{2} \Rightarrow e^{2}=1-e^{2} \quad\left[\because \frac{b}{a}=e\right]$</span></p>
<p><span class="math-tex">$\Rightarrow$</span> 2e<sup>2</sup> = 1<br />
<span class="math-tex">$\Rightarrow \quad e=\frac{1}{\sqrt{2}}$</span> ...(ii)<br />
From Eqs. (i) and (ii), we get<br />
<span class="math-tex">$b=4 \sqrt{2}$</span><br />
Now, <span class="math-tex">$a^{2}=\frac{b^{2}}{1-e^{2}}=\frac{32}{1-\frac{1}{2}}=64$</span><br />
<span class="math-tex">$\therefore$</span> Equation of ellipse be <span class="math-tex">$\frac{x^{2}}{64}+\frac{y^{2}}{32}=1$</span><br />
Now, check all the options.<br />
Only <span class="math-tex">$(4 \sqrt{3}, 2 \sqrt{2})$</span>, satisfy the above equation.</p>
Correct Answer: C