Question:
<p>The equation of parabola whose focus is (5, 3) and directrix is 3x - 4y + 1 = 0, is</p>
<p style="display:inline">(3x + 4y)<sup>2</sup> - 142x - 256y + 849 = 0</p>
<p style="display:inline">(4x + 3y)<sup>2</sup> - 256x - 142y + 849 = 0</p>
<p style="display:inline">(3x - 4y)<sup>2</sup> - 256x - 142y + 849 = 0</p>
<p style="display:inline">(4x - 3y)<sup>2</sup> - 256x - 142y + 849 = 0</p>
Step-by-Step Solution
Key Concept: A parabola is defined as the locus of a point whose distance from the focus is equal to its perpendicular distance from the directrix.
<p>Let P(x, y) be any point on the parabola.<br />
Focus S = (5, 3)<br />
<span class="math-tex">\(\Rightarrow\)</span> (x - 5)<sup>2</sup> + (y - 3)<sup>2</sup> = <span class="math-tex">\(\left(\frac{3 x-4 y+1}{\sqrt{9+16}}\right)^{2}\)</span><br />
<span class="math-tex">\(\Rightarrow\)</span> 25(x<sup>2</sup> + 25 - 10x + y<sup>2</sup> + 9 - 6y)<br />
= 9x<sup>2</sup> + 16y<sup>2</sup> + 1 - 12xy + 6x - 8y - 12xy<br />
<span class="math-tex">\(\Rightarrow\)</span> 16x<sup>2</sup> + 9y<sup>2</sup> - 256x - 142y + 24xy + 849 = 0<br />
<span class="math-tex">\(\Rightarrow\)</span> (4x + 3y)<sup>2</sup> - 256x - 142y + 849 = 0</p>
Correct Answer: B