Question:
<p>The locus of a point which divides the line segment joining the point (0, -1) and a point on the parabola, x<sup>2</sup> = 4y, internally in the ratio 1 : 2, is:</p>
<p style="display:inline">x<sup>2</sup> - 3y = 2</p>
<p style="display:inline">9x<sup>2</sup> - 3y = 2</p>
<p style="display:inline">4x<sup>2</sup> - 3y = 2</p>
<p style="display:inline">9x<sup>2</sup> - 12y = 8</p>
Step-by-Step Solution
Key Concept: Represent the point on the parabola parametrically and apply the section formula to relate the locus point's coordinates to the parameter, which is then eliminated.
<p>Let point P be (2t, t<sup>2</sup>) and Q be (h, k)<br />
Using section formula,<br />
<span class="math-tex">$h=\frac{2 t}{3}, k=\frac{-2+t^{2}}{3}$</span><br />
Hence, locus is 3k + 2 <span class="math-tex">$=\left(\frac{3 h}{2}\right)^{2}$</span><br />
<span class="math-tex">$\Rightarrow$</span> 9x<sup>2</sup> - 12y + 8</p>
Correct Answer: D