how to know when to use family of straight lines

An equation of the type ax2+ 2hxy + by2which is a homogeneous equation of degree 2 denotes a pair of straight lines passing through the origin. The lines may be real, coincident or imaginary depending on the conditions satisfied by them:

If h2> ab, then the lines are real and distinct.

If h2< ab, then the lines are imaginary with the point of intersection as (0, 0).

If h2= ab, then the lines are coincident.

Ifα is the acute angle between the pair of straight lines, then tanα = |2√(h2-ab)/(a + b)|.

These lines are perpendicular to each other if the coefficient of x2+ coefficient of y2= 0.

If the coefficient of xy = 0, then the lines are equally inclined to the x-axis.

The lines are said to be coincident if h2= ab.





What do you mean by the joint equation of a pair of straight lines?

If the line lx + my + n = 0, (n ≠ 0) i.e. the line not passing through origin) cuts the curve ax2+ by2+ 2gx + 2fy + c = 0 at two points A and B, then the joint equation of straight lines passing through A and B and the origin is given by homogenizing the equation of the curve by the equation of the line. i.e. if we have a line lx + my + n = 0 and a second degree curve given by the equation ax2+ 2hxy + by2+ 2gx + 2fy + c = 0, then the joint equation of a pair of straight lines joining origin to the points of intersection of both is given by

ax2+ 2hxy + by2+ (2gx + 2fy) (lx+my/–n) + c (lx+my/–n)2= 0. Thisis the equation of the lines OA and OB.



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