Condition for lx+my+n=0 to be tangent to ellipse

The line lx + my + n = 0 touches the ellipse x²/a² + y²/b² = 1 if it meets the curve in two coincident points.

Method 1: using y = mx + c

For the ellipse, the line y = Mx + c is a tangent iff c² = a²M² + b².

Write lx + my + n = 0 as y = −(l/m)x − n/m. Here slope M = −l/m and c = −n/m. The condition becomes

n²/m² = a² l²/m² + b²

a²l² + b²m² = n²

Method 2: direct substitution (coincident roots)

Take y = −(lx + n)/m and put in b²x² + a²y² = a²b²:

b²m²x² + a²(lx + n)² = a²b²m²

(b²m² + a²l²)x² + 2a²ln x + a²(n² − b²m²) = 0

For equal roots, the discriminant is zero:

4a⁴l²n² − 4(b²m² + a²l²)a²(n² − b²m²) = 0

Divide by 4a²:

a²l²n² − (b²m²n² − b⁴m⁴ + a²l²n² − a²b²l²m²) = 0 b²m²(b²m² + a²l² − n²) = 0

so a²l² + b²m² = n².

Point of contact

The point of contact is (−a²l/n, −b²m/n).