Theorem
The condition for a straight line y = mx + c to be a tangent to the ellipse
is c
2
= a
2
m
2
+ b
2
.
Proof:
Given, y = mx + c ----- (i) and
----- (ii)
Substituting (i) in (ii), we get
(a
2
m
2
+ b
2
) x
2
+ 2a
2
mcx + a
2
(c
2
– b
2
) = 0 ...... (iii)
The line will touch the ellipse iff the two points are coincident.
⇔ discriminant of (iii) is zero.
⇔ 4a
4
c
2
m
2
– 4(a
2
m
2
+ b
2
) a
2
(c
2
– b
2
) = 0
⇔ c
2
= a
2
m
2
+ b
2
⇔ c = ±