Hyperbola
Tangent to Rectangular Hyperbola
Grade 11
Question:
<p>Straight line \(Ax + By + D = 0\) would be tangent to \(xy = c^2\), if:</p>
<p>(a) \(A > 0, B > 0\)</p>
<p>(b) \(A < 0, B < 0\)</p>
<p>(c) \(A > 0, B < 0\)</p>
<p>(d) \(A < 0, B > 0\)</p>
Step-by-Step Solution
Key Concept: For tangency to the rectangular hyperbola, the discriminant of the resulting quadratic must be zero, which constrains the sign relationship between coefficients A and B.
<p>For the line \(Ax + By + D = 0\) to be tangent to the rectangular hyperbola \(xy = c^2\), we substitute \(y = -\frac{Ax + D}{B}\) into \(xy = c^2\), giving \(Ax^2 + Dx - \frac{B c^2}{1} = 0\). For tangency, the discriminant must be zero: \(D^2 + 4ABc^2 = 0\), which requires \(AB < 0\). This means A and B have opposite signs.</p>
Correct Answer: b, d