<p>Consider an ellipse \(\frac{x^2}{36} + \frac{y^2}{18} = 1\). There is a hyperbola whose one asymptote is the major axis of the given ellipse. If eccentricity of the given ellipse and hyperbola are reciprocal to each other, both have the same centre and both touch each other in the first and third quadrants. <strong>Find the number of points in the x-y plane from where perpendicular tangents can be drawn to the hyperbola.</strong></p>
Step-by-Step Solution
Key Concept: The director circle of a hyperbola is imaginary when \(a^2 - b^2 < 0\), meaning no perpendicular tangents exist from any real point.
<p><strong>Solution approach:</strong> For a hyperbola, perpendicular tangents can be drawn from points on the auxiliary circle (the director circle). The director circle of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is given by \(x^2 + y^2 = a^2 - b^2\). For the hyperbola in this problem, this radius-squared becomes negative, indicating that the director circle is imaginary. Therefore, no real points exist from which perpendicular tangents can be drawn.</p>
Correct Answer: A