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Point of Intersection of an Ellipse with a Line

​
ellipse
semiaxes
a
0.6
b
0.45
line
slope angle
ϕ
0.45
x intercept
c
0.2
zoom
1
This Demonstration shows a ruler and compass construction of an intersection of an ellipse with semiaxes
a
and
b
with a line.
Suppose the ellipse (in blue) has the equation
2
x
2
a
+
2
y
2
b
=1
and that the line (in red) has the equation
y=k(x-c)
.
The linear transformation
(x,y)(x,by/a)
transforms the ellipse into the circle
2
x
+
2
y
=
2
a
and the line into a new line with the equation
y=ak/b(x-c)
. We can construct this line with slope
ak/b
and its intersection with the circle. Let
P(x,y)
be one of the intersection points. Then
R(x,by/a)
is one of the intersections of the ellipse with the red line. To construct the point
R
, we must find the intersection point
E
of the line
CP
and the line
AB
. The point
R
is the intersection of the line
DE
and the perpendicular line at
P
on
AB
.
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