Question:

<p><span class="math-tex">\(T=S_{1}\)</span><br /> <span class="math-tex">\(\Rightarrow \frac{x \cdot 1}{4}+\frac{y \cdot 1 / 2}{2}=\frac{1}{4}+\frac{1}{8} \Rightarrow x+y=\frac{3}{2}\)</span><br /> solving with ellipse <span class="math-tex">\(x^{2}+2 y^{2}=4\)</span>, we get<br /> <span class="math-tex">\(x^{2}+2 y^{2}=4\)</span><br /> <span class="math-tex">\(\Rightarrow x^{2}+2\left(\frac{3}{2}-x\right)^{2}=4\)</span><br /> <span class="math-tex">\(\Rightarrow 6 x^{2}-12 x+1=0\)</span><br /> <span class="math-tex">\(\Rightarrow x_{1}+x_{2}=2\)</span><br /> <span class="math-tex">\(\Rightarrow x_{1} x_{2}=\frac{1}{6}\)</span><br /> <span class="math-tex">\(\therefore\left|x_{2}-x_{1}\right|=\sqrt{\left(x_{2}+x_{1}\right)^{2}-4 x_{1} x_{2}}\)</span><br /> <span class="math-tex">\(=\sqrt{4-4 / 6}\)</span><br /> also, <span class="math-tex">\(y_{2}=\frac{3}{2}-x_{2}\)</span><br /> <span class="math-tex">\(y_{1}=\frac{3}{2}-x_{1}\)</span><br /> <span class="math-tex">\(\therefore y_{2}-y_{1}=x_{2}-x_{1}\)</span><br /> length of the chord of the ellipse, <span class="math-tex">\(P R\)</span><br /> <span class="math-tex">\(=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)</span><br /> <span class="math-tex">\(\Rightarrow P R=\sqrt{2} \cdot 2 \cdot \frac{\sqrt{5}}{\sqrt{2} \sqrt{3}}=\frac{2}{3} \sqrt{15}\)</span></p>