Hyperbola
Common Tangents to Two Hyperbolas
Grade 11
Question:
<p>The equation(s) to common tangent(s) to the two hyperbolas \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) and \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) is/are:</p>
<p>(a) \(y = x + \sqrt{a^2 - b^2}\)</p>
<p>(b) \(y = x - \sqrt{a^2 - b^2}\)</p>
<p>(c) \(y = -x + \sqrt{a^2 - b^2}\)</p>
<p>(d) \(y = -x - \sqrt{a^2 - b^2}\)</p>
Step-by-Step Solution
Key Concept: For a line to be tangent to both hyperbolas, it must satisfy the tangency conditions for both curves simultaneously, leading to a system of equations.
<p>A common tangent to both hyperbolas must satisfy the tangency condition for each. For the first hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), a line \(y = mx + c\) is tangent if \(c^2 = a^2m^2 - b^2\). For the second hyperbola \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the tangency condition is \(c^2 = a^2 - b^2m^2\). Solving these simultaneously gives \(m = \pm 1\) and \(c = \pm\sqrt{a^2 - b^2}\).</p>
Correct Answer: a, b, c, d