Question:
<p>The distance between the vertex and the focus of the parabola x<sup>2</sup> - 2x + 3y - 2 = 0 is</p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{4}{5}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{3}{4}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{5}{6}\)</span></p>
Step-by-Step Solution
Key Concept: Complete the square to express the parabola in the standard form (x-h)² = ±4a(y-k), where the distance from the vertex to the focus is simply the value of a.
<p>Given equation of parabola<br />
x<sup>2</sup> - 2x + 3y - 2 = 0<br />
<span class="math-tex">$\Leftrightarrow$</span> (x - 1)<sup>2</sup> = -3(y - 1)<br />
<span class="math-tex">$\Rightarrow$</span> vertex = (1, 1), focus <span class="math-tex">$=\left(1, \frac{1}{4}\right)$</span><br />
<span class="math-tex">$\Rightarrow$</span> Both the points lie on x = 1<br />
<span class="math-tex">$\Rightarrow$</span> The distance between focus and vertex<br />
<span class="math-tex">$=\left|1-\frac{1}{4}\right|=\frac{3}{4}$</span></p>
Correct Answer: C