Ellipse
Tangents to Ellipse and Hyperbola
Grade 11
Question:
<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 equation of the common tangent to the given ellipse and hyperbola in the first quadrant.</strong></p>
<p>(a) \(\frac{x}{2} + y = 3\)</p>
<p>(b) \(\frac{x}{2} + y = 6\)</p>
<p>(c) \(x + y\sqrt{2} = 3\sqrt{2}\)</p>
<p>(d) \(x + y\sqrt{2} = 6\)</p>
Step-by-Step Solution
Key Concept: A common tangent must be tangent to both conics simultaneously; use tangency conditions (discriminant = 0) for both curves.
<p><strong>Solution approach:</strong> A common tangent to both the ellipse and hyperbola must satisfy the tangency conditions for both curves. For the ellipse \(\frac{x^2}{36} + \frac{y^2}{18} = 1\) and the hyperbola (determined from previous conditions), substitute the line equation \(y = mx + c\) into both conic equations and apply the condition that the discriminant equals zero. Solving these simultaneously and restricting to the first quadrant gives the common tangent equation \(\frac{x}{2} + y = 3\).</p>
Correct Answer: A