Question:
<p>The equation of the parabola whose vertex is at (2, -1) and focus at (2, -3) is</p>
<p style="display:inline">x<sup>2</sup> - 4x + 8y + 12 = 0</p>
<p style="display:inline">x<sup>2</sup> - 4x + 12 = 0</p>
<p style="display:inline">x<sup>2</sup> + 4x - 8y - 12 = 0</p>
<p style="display:inline"><img alt="" height="143" src="https://i.imgur.com/3utv1eb.png" width="147"/><br/>
a = |VS|<br/>
<span class="math-tex">\(=\sqrt{(2-2)^{2}+(-3+1)^{2}}\)</span><br/>
= 2<br/>
The required equation is<br/>
(x - 2)<sup>2</sup> = - 4 <span class="math-tex">\(\times\)</span> 2 (y + 1)<br/>
<span class="math-tex">\(\Rightarrow\)</span> x<sup>2</sup> + 4 - 4x = -8y - 8<br/>
<span class="math-tex">\(\Rightarrow\)</span> x<sup>2</sup> - 4x + 8y + 12 = 0</p>
Step-by-Step Solution
Key Concept: Identify the orientation and focal length from the vertex and focus to apply the standard translated equation (x-h)² = -4a(y-k).
<p>x<sup>2</sup> - 4x + 8y + 12 = 0</p>
Correct Answer: A