<p>The locus of a point \(P(\alpha, \beta)\) moving under the condition that the line \(y = \alpha x + \beta\) is a tangent to the hyperbola \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\) is</p>
Step-by-Step Solution
Key Concept: A line y = mx + c is tangent to the hyperbola x²/a² - y²/b² = 1 if and only if c² = a²m² - b². Since the line y = αx + β is tangent, we have β² = a²α² - b², which is the locus equation in terms of α and β.
<p><strong>Step 1:</strong> Recall the condition for a line y = mx + c to be tangent to the hyperbola x²/a² - y²/b² = 1.</p><p>Substituting y = mx + c into the hyperbola equation:</p><p>x²/a² - (mx + c)²/b² = 1</p><p><strong>Step 2:</strong> For tangency, the discriminant of the resulting quadratic in x must equal zero. This gives the tangency condition: <strong>c² = a²m² - b²</strong></p><p><strong>Step 3:</strong> In our problem, the line is y = αx + β, so comparing with y = mx + c, we have m = α and c = β.</p><p><strong>Step 4:</strong> Applying the tangency condition with m = α and c = β:</p><p>β² = a²α² - b²</p><p>Rearranging: <strong>a²α² - β² = b²</strong></p><p>Or equivalently: <strong>α²/b² - β²/b⁴ · a² = 1</strong> (in standard form)</p><p>This is a hyperbola in the (α, β) coordinate system.</p><p>∴ Answer: D</p>
Correct Answer: D