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in Mathematics by (53.3k points)

Suppose that the straight line lx + my = 1 meets the curve represented by the second-degree general equation S ≡ ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 at two point A and B. If O is the origin, the combined equation of the pair of lines bar(OA) and (bar) OB is

S' ≡ ax2 + 2hxy + by2 + 2(gx + fy)(lx + my) + c (lx + my)2 = 0

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Clearly the coordinates of both points A and B satisfy the line equation lx + my =1 as well as S = 0 and hence points A and B satisfy S' = 0. Also (0, 0) satisfies S' = 0. That is, S' = 0 passes through A, B and origin (see Fig.). On simplification, we can see that S' = 0 is a homogeneous equation of second degree representing pair of lines, which are nothing but the lines (bar)OA and (bar)OB.

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