Question:
<p>The locus of the middle points of the focal chords of the parabola, y<sup>2</sup> = 4x is:</p>
<p style="display:inline">y<sup>2</sup> = 2(1 - x)</p>
<p style="display:inline">y<sup>2</sup> = 3(x - 1)</p>
<p style="display:inline">y<sup>2</sup> = x - 1</p>
<p style="display:inline">y<sup>2</sup> = 2(x - 1)</p>
Step-by-Step Solution
Key Concept: Represent the midpoint's coordinates using the parametric endpoints of a focal chord and eliminate the parameter $t$ using the algebraic relationship between $(t-1/t)$ and $(t^2+1/t^2)$.
<p>We have <br />
y<sup>2 </sup>= 4x<br />
We know that ends of focal chords are (at<sup>2</sup>, 2at) and <span class="math-tex">\(\left(\frac{a}{t^{2}},-\frac{2 a}{t}\right)\)</span><br />
here a = 1<br />
Let (h, k) be the mid point of the chord.<br />
<span class="math-tex">\(\Rightarrow {k}=\frac{2 {t}+\left(-\frac{2}{t}\right)}{2}\)</span><br />
<span class="math-tex">\(\Rightarrow 2 {k}=2 {t}-\frac{2}{{t}}\)</span><br />
<span class="math-tex">\(\Rightarrow {k}={t}-\frac{{1}}{{t}}\)</span> ...(i)<br />
h = <span class="math-tex">\(\frac{\mathbf{t}^{2}+\frac{{1}}{{t}^{2}}}{{2}}\)</span><br />
<span class="math-tex">\(\Rightarrow 2 h=\left(t-\frac{1}{t}\right)^{2}\)</span> + 2<br />
<span class="math-tex">\(\Rightarrow\)</span> 2h = k<sup>2</sup> + 2<br />
To get equation of locus replace<br />
h <span class="math-tex">\(\rightarrow\)</span> x and k <span class="math-tex">\(\rightarrow\)</span> y<br />
2x = y<sup>2</sup> + 2<br />
y<sup>2 </sup>= 2(x − 1)</p>
Correct Answer: D