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

Use the parametric point P(a cosφ, b sinφ) on the ellipse x²/a² + y²/b² = 1.

Normal at P

The tangent at P is x cosφ/a + y sinφ/b = 1, with slope −(b cosφ)/(a sinφ). The normal is perpendicular to it and passes through P:

ax/cosφ − by/sinφ = a² − b²

Comparing with lx + my + n = 0

Write the given line as lx + my = −n. Two equations represent the same line if their coefficients are proportional:

(a/cosφ)/l = (−b/sinφ)/m = (a² − b²)/(−n)

From this:

cosφ = −an / [l(a² − b²)] sinφ = bn / [m(a² − b²)]

Eliminate φ

Using cos²φ + sin²φ = 1:

a²n²/[l²(a² − b²)²] + b²n²/[m²(a² − b²)²] = 1

Multiplying by (a² − b²)²/n²:

a²/l² + b²/m² = (a² − b²)²/n²

This is the required condition for lx + my + n = 0 to be a normal to the ellipse.