Condition for x cosα + y sinα = p to touch ellipse

The line x cosα + y sinα = p is in the normal (perpendicular) form, with p the distance of the line from the origin.

Using the tangent condition

Write it as x cosα + y sinα − p = 0, so l = cosα, m = sinα, n = −p. The line lx + my + n = 0 touches x²/a² + y²/b² = 1 if n² = a²l² + b²m² (shown below), hence

p² = a² cos²α + b² sin²α

Proof of the condition

Write the line as y = (p − x cosα)/sinα and put in b²x² + a²y² = a²b²:

b²x² sin²α + a²(p − x cosα)² = a²b² sin²α

(b² sin²α + a² cos²α)x² − 2a²p cosα · x + a²(p² − b² sin²α) = 0

A tangent gives equal roots (discriminant = 0):

4a⁴p² cos²α = 4(b² sin²α + a² cos²α)·a²(p² − b² sin²α)

a²p² cos²α = (b² sin²α + a² cos²α)p² − b² sin²α (b² sin²α + a² cos²α)

0 = b² sin²α · p² − b² sin²α (b² sin²α + a² cos²α)

p² = a² cos²α + b² sin²α

Hence the line x cosα + y sinα = p touches the ellipse iff p² = a²cos²α + b²sin²α. (Check: for α = 0 the line x = p touches the ellipse at x = ±a, giving p² = a² ✓.)