Question:
<p>For the ellipse 4(x - 2y + 1)<sup>2</sup> + 9(2x + y + 2)<sup>2</sup> = 180, lengths of major and minor axes are respectively</p>
<p style="display:inline">4 and 6</p>
<p style="display:inline"><span class="math-tex">\(6 \sqrt{5}\)</span> and <span class="math-tex">\(4 \sqrt{5}\)</span></p>
<p style="display:inline">6 and 4</p>
<p style="display:inline"><span class="math-tex">\(4 \sqrt{5}\)</span> and <span class="math-tex">\(6 \sqrt{5}\)</span></p>
Step-by-Step Solution
Key Concept: Convert the given equation into standard form by normalizing the linear terms to represent perpendicular distances from the ellipse's axes.
<p>We have, 4(x - 2y + 1)<sup>2</sup> + 9(2x + y + 2)<sup>2</sup> = 180<br />
<span class="math-tex">\(\Rightarrow \frac{(x-2 y+1)^{2}}{45}+\frac{(2 x+y+2)^{2}}{20}=1\)</span><br />
<span class="math-tex">\(\Rightarrow \frac{\left(\frac{x-2 y+1}{\sqrt{1+4}}\right)^{2}}{3^{2}}+\frac{\left(\frac{2 x+y+2}{\sqrt{4+1}}\right)^{2}}{2^{2}}=1\)</span><br />
<span class="math-tex">\(\Rightarrow \frac{X^{2}}{3^{2}}+\frac{Y^{2}}{2^{2}}=1\)</span>, where<br />
<span class="math-tex">\(\mathrm{X}=\frac{x-2 y+1}{\sqrt{5}}\)</span> and <span class="math-tex">\(\mathrm{Y}=\frac{2 x+y+2}{\sqrt{5}}\)</span><br />
<span class="math-tex">\(\Rightarrow\)</span> Length of the major axis = 2(3) = 6,<br />
Length of the minor axis = 2(2) = 4.</p>
Correct Answer: C