Question:
<p>The one which does not represent a hyperbola is:</p>
<p style="display:inline">(x - 1)(y - 3) = 3</p>
<p style="display:inline">x<sup>2</sup> - y<sup>2</sup> = 5</p>
<p style="display:inline">xy = 1</p>
<p style="display:inline">x<sup>2</sup> - y<sup>2</sup> = 0</p>
Step-by-Step Solution
Key Concept: A hyperbola requires the discriminant condition h² - ab > 0 for the general conic ax² + 2hxy + by² + 2gx + 2fy + c = 0. For option D: x² - y² = 0, we have a=1, h=0, b=-1, giving h² - ab = 0 - (1)(-1) = -1 < 0, which represents a pair of intersecting straight lines, not a hyperbola.
<p>For hyperbola <span class="math-tex">$\Delta \ne$</span> 0 and h<sup>2</sup> > ab<br />
Here <span class="math-tex">$\Delta$</span> = 0</p>
Correct Answer: D