Theorem

The condition for a straight line y = mx + c to be a tangent to the ellipse

is c² = a²m² + b².

Proof:

Given, y = mx + c ----- (i) and

----- (ii)

Substituting (i) in (ii), we get

(a²m² + b²) x² + 2a² mcx + a² (c² – b²) = 0 ...... (iii)

The line will touch the ellipse iff the two points are coincident.

⇔ discriminant of (iii) is zero.

⇔ 4a⁴ c²m² – 4(a²m² + b²) a² (c² – b²) = 0

⇔ c² = a²m² + b²

⇔ c = ±