
Applications of the Derivative to Geometry 6-31
The pedal equation of a curve is usually written as
= = =( ) ( ) ( , ) .or or 0
Case 1: To form a pedal equation whose Cartesian equation is given. Let
( , ) = 0 (6.8)
be the given Cartesian equation of the curve.
The equation of the tangent at the point (x, y) is given by
Y y
dy
dx
X x
Y
dy
dx
X x
dy
dx
y
− = −
− + − =
( )
.Or 0
If p is the length of the perpendicular from (0, 0) to the above tangent, then
∴ =
−
+
p
x
dy
dx
y
dy
dx
1
2
1 2/
. (6.9)
Also,
r OP x y
2 2 2 2
= = + . (6.10)
Eliminating x and y from Eqs. (6.8)–(6.10), we get the pedal equation of the given curve. ...