Secants are drawn from the point P(-1, 3) to the curve x² + y² - 2x + 4y - 8 = 0, which meets the circle at A and B. If the minimum values of PA + PB is 'K', then what is length of the focal chord to the parabola y² = 4Kx?

The minimum value of PA + PB occurs when the secant line passes through the center of the circle at (1, -2), meaning it equals the length of the chord through P. The distance from P(-1, 3) to the center (1, -2) is sqrt((-1 - 1)^2 + (3 - (-2))^2) = sqrt(4 + 25) = sqrt(29), and the circle's radius is sqrt(1 + 4 + 8) = sqrt(13), making the chord length 2 * sqrt(29 - 13) = 2 * 4 = 8, so K = 8. The length of the focal chord (latus rectum) of the parabola y^2 = 4ax is 4a, and here 4a = 4K = 4 * 8 = 32. The focal chord length is 32.

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