<p>Tangents are drawn from the point \((-1, 2)\) on the parabola \(y^2 = 4x\). The length these tangents will intercept on the line \(x = 2\) is:</p>
Step-by-Step Solution
Key Concept: Find the two tangent lines from external point, substitute x = 2 to get points on directrix line, calculate distance between intersection points.
<p>For parabola \(y^2 = 4x\), tangent at parameter \(t\) is \(y = \frac{x}{t} + t\).</p><p>If tangent passes through \((-1, 2)\): \(2 = \frac{-1}{t} + t\), giving \(t^2 - 2t - 1 = 0\)</p><p>\(t = 1 \pm \sqrt{2}\)</p><p>At \(x = 2\): \(y_1 = \frac{2}{1+\sqrt{2}} + (1+\sqrt{2})\) and \(y_2 = \frac{2}{1-\sqrt{2}} + (1-\sqrt{2})\)</p><p>Length of intercept = \(|y_1 - y_2| = 6\)</p>
Correct Answer: a