Applications of Derivatives
Normal to Curves
Grade 12
Question:
<p>If the line \(ax + by + c = 0\) is a normal to the curve \(xy = 1\), then</p>
<p>(a) \(a > 0, b > 0\)</p>
<p>(b) \(a > 0, b < 0\)</p>
<p>(c) \(a < 0, b < 0\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Find the general form of the normal to the hyperbola and observe that signs depend on the parameter value.
<p>For the curve $xy = 1$, we have $y = \frac{1}{x}$.</p><p>At point $(t, \frac{1}{t})$, the slope is $\frac{dy}{dx} = -\frac{1}{t^2}$.</p><p>The normal has slope $t^2$.</p><p>Normal equation: $y - \frac{1}{t} = t^2(x - t)$, which simplifies to $t^2 x - y - t^3 + \frac{1}{t} = 0$.</p><p>The signs of coefficients depend on the value of $t$, so no fixed sign relationship exists for all normals.</p>
Correct Answer: D