7
$\begingroup$

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" the line with the curve . They basically said that the required curve would be $$ 2x^2 +3y^2 -5\left(\frac{3x-4y}{5}\right)^2=0$$ What is this ? And how does this give required curve??

$\endgroup$
  • 1
    $\begingroup$ What are the properties of wanted curve? $\endgroup$ – science Mar 5 '15 at 6:51
  • 2
    $\begingroup$ I've been told this is applicable to all curves $\endgroup$ – Tesla Mar 5 '15 at 7:00
15
$\begingroup$

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:

enter image description here

I am given a curve of the form $ax^2+by^2+cxy+dy+ex+f=0$ 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, $\dfrac{px+qy}{r}=1,$ I homogenize the curve in this manner: $$ax^2+by^2+cxy+dy\left(\frac{px+qy}{r}\right)+ex\left(\frac{px+qy}{r}\right)+f\left(\frac{px+qy}{r}\right)^2=0\tag{i}$$ To justify that (i) is, in fact, the joint equation of $OA$ and $OB$, I give two reasons:

  1. This is a homogeneous equation, so it is for sure a pair of straight lines through the origin.
  2. It passes through the points $A$ and $B$ as they are the points of intersection of the given curve and line, so both the relations $\dfrac{px+qy}{r}=1$ and $ax^2+by^2+cxy+dy+ex+f=0$ are satisfied.

With a similar argument we can say that this 'trick' of homogenization can be applied to any curve of any degree but I think a rigorous proof would require knowledge of projective geometry,scale invariance of homogeneous curves etc.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.