Line Through Two Points In The Plane
Math 98
Lines
It is easy to write the equation of the unique line that connects two points
in the plane, but you have to do it right. Hereโs two ways (even though they
are essentially the same).
If you have two points in the plane, say (a, b), and (c, d), it is easy, for
example from a picture, to find the slope of the line connecting them: indeed,
that has to be
m=dโb
cโa=bโd
aโc
Since a line has an equation of the form
y=mx +k
we only need to find k, and that we can do by โforcingโ the line to go through
one of the point (the pother one will also follow, since we have picked the right
slope. For example, we can plug in x=a, y =b, and see that kmust be such
that
b=ma +k=bโd
aโca+k
k=bโabโd
aโc=b(aโc)โa(bโd)
aโc=ab โbc โab +ad
aโc=ad โbc
aโc
and the equation now becomes
y=mx +k=bโd
aโcx+ad โbc
aโc=(bโd)x+ (ad โbc)
aโc(1)
You really would not want to memorize (1). It is much faster to retrace the
steps we made to find k!
Another way is to write a linear function that will become an identity when
we plug in the two pairs describing the points. One standard way to do this
is to make sure we get 0 = 0 when, say, x=a, y =b. This means that the
function must look like
A(yโb) = B(xโa)
for some numbers Aand B. Now, we make sure that the two sides are both
equal to, say 1, when x=c, y =d. I.e., we need
A(dโb)=1, B (cโa) = 1
1
