Question

In what type of questions we do homogenization of curves

Answer 1

A homogeneous equation, ie, an equation with all terms of same degree, will always represent a set of straight lines passing through the origin. With this fact in mind, consider the following example:

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I am given a curve of the form ax2+by2+cxy+dy+ex+f=0 and the line px+qy=r which intersects it at points A and B. I need to find the joint equation of the lines OA and OB.

To achieve this, I homogenize the equation of the curve by multiplying any term of less than second degree with a factor that doesn't change anything except the degree, ie , unity. Since for any point on the given line, px+qyr=1, I homogenize the curve in this manner:

ax2+by2+cxy+dy(px+qyr)+ex(px+qyr)+f(px+qyr)2=0(i)