
Applications of the Derivative to Geometry 6-21
Hence, we have m
1
×m
2
= -1.
⇒ −
⋅−
= −
⇒ =− ⇒ = −
xa
yb
xc
yd
acx bdy
ac
bd
y
1
2 2
22
2
.
We have from the given equations
ax by cx dy
a c x b d y
y
x
a c
b d
2 2 2 2
2 2
2
2
+ = +
⇒ − =− −
⇒ − =
−
−
( ) ( )
.
Thus,
ac
bd
a c
b d
=
−
−
.
Therefore, the required condition is
ac
bd
a c
b d
=
−
−
.
6.2.4 Lengths of Tangent, Normal, Sub-tangent and Sub-normal
The length of the tangent is defined to be the portion of the tangent line between the point P
1
(x
1
, y
1
) and
the point where the tangent crosses the x-axis. In Fig. 6.10, the length of the tangent is the line MP
1
.
Similarly, we can state that the length of the normal ...