It is known that the straight line L: y = KX + m intersecting parabola C: x ^ 2 = 4Y is tangent to two different points a and B respectively Let two tangents intersect at point M. if M (2, - 1), the equation of line L is obtained | Math enthusiast | Mathematical art

It is known that the straight line L: y = KX + m intersecting parabola C: x ^ 2 = 4Y is tangent to two different points a and B respectively Let two tangents intersect at point M. if M (2, - 1), the equation of line L is obtained

Let y + 1 = a (X-2) y = a (X-2) - 1, substitute x ^ 2 = 4yx ^ 2-4ax + 8A + 4 = 016a ^ 2-4 (8a + 4) = 0A = 1 + (radical 2), or 1 - (radical 2) x = 4A / 2 = 2A = 2 + 2 (radical 2), or 2-2 (radical 2) so: the coordinates of a and B (2 + 2 (radical 2), 3 + 2 (radical 2)), (2-2 (radical 2), 3-2 (radical 2)) so

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1. X ^ 2 = 4Y, the tangent of line L passing through a and B is L1 and L2 (1) Verification of L1 vertical L2 (2) It is proved that the focus of L1 and L2 is on the directrix
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