Question

Find the equation to the two straight lines joining the origin to the points in which the straight line Ix + my = n, meets the locus whose equation is ax² + 2hxy + by² + 2gx + 2fy + c = 0

Answer 1

Ix + my = n..........(1)

ax² + 2hxy + by² + 2gx + 2fy + c = 0.......(2)

The equation (1) may be written

The coordinates of the points in which the straight line meets the locus satisfy both equation (2) and equation (3), and hence satisfy the equation

[For at the points where (3) and (4) are true it is clear that (2) is true.]

Hence (4) represents some locus which passes through the intersections of (2) and (3).

But, since the equation (4) is homogeneous and of the second degree, it represents two straight lines passing through the origin.

It therefore must represent the two straight lines joining the origin to the intersections of (2) and (3).