Question:
<p>If <span class="math-tex">\(\alpha x+\beta y=109\)</span> is the equation of the chord of the ellipse <span class="math-tex">\(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\)</span>, whose mid point is <span class="math-tex">\(\left(\frac{5}{2}, \frac{1}{2}\right)\)</span>, then <span class="math-tex">\(\alpha+\beta\)</span> is equal to:</p>
<p style="display:inline">72</p>
<p style="display:inline">58</p>
<p style="display:inline">37</p>
<p style="display:inline">46</p>
Step-by-Step Solution
Key Concept: The equation of a chord of an ellipse with a given midpoint $(x_1, y_1)$ is obtained using the formula $T = S_1$.
<p><img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1776056789-trc9ka.jpg" style="height:148px; width:200px" /><br />
Equation of chord <span class="math-tex">${T}={S}_{1}$</span><br />
<span class="math-tex">$\frac{5}{2}\left(\frac{x}{9}\right)+\frac{1}{2}\left(\frac{y}{4}\right)=\frac{25}{36}+\frac{1}{16}$</span><br />
<span class="math-tex">$\Rightarrow \frac{5 x}{18}+\frac{y}{8}=\frac{109}{144}$</span><br />
<span class="math-tex">$\Rightarrow 40 x+18 y=109$</span><br />
<span class="math-tex">$\Rightarrow \alpha=40, \beta=18$</span><br />
<span class="math-tex">$\Rightarrow \alpha+\beta=40+18=58$</span></p>
Correct Answer: B