Question:
<p>The equation of the chord, of the ellipse <span class="math-tex">\(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)</span>, whose midpoint is <span class="math-tex">\((3,1)\)</span> is:</p>
<p style="display:inline"><span class="math-tex">\(4 x+122 y=134\)</span></p>
<p style="display:inline"><span class="math-tex">\(48 x+25 y=169\)</span></p>
<p style="display:inline"><span class="math-tex">\(25 x+101 y=176\)</span></p>
<p style="display:inline"><span class="math-tex">\(5 x+16 y=31\)</span></p>
Step-by-Step Solution
Key Concept: The equation of a chord of a conic section with a given midpoint $(x_1, y_1)$ is determined by the formula $T = S_1$, where $T$ is the tangent expression and $S_1$ is the power of the point.
<p>Equation of chord with given middle point <span class="math-tex">$T=S_{1}$</span><br />
<span class="math-tex">$\Rightarrow \frac{3 x}{25}+\frac{y}{16}-1=\frac{9}{25}+\frac{1}{16}-1$</span><br />
<span class="math-tex">$\Rightarrow 48 x+25 y=144+25$</span><br />
<span class="math-tex">$\therefore 48 x+25 y=169$</span></p>
Correct Answer: B