Tangent Lines to an Ellipse

Question

Determine the equations of the tangent lines to the ellipse

(E) b²x² +a²y² = a²b²

that have the given slope m.

Answer 1

2x/a² + 2yy'/b² = 0; y' = - xb²/(ya²) = m; y = - xb²/(ma²);

x²/a² + x²b²/(m²a⁴) = 1; x²/a²(1 + b²/(m²a²)) =1 ; x = ± ma²/√(m²a² + b²); y = - + b²/√(m²a² + b²)

2 tangent lines: y = - b²/√(m²a² + b²) + m(x - ma²/√(m²a² + b²));

y = b²/√(m²a² + b²) + m(x + ma²/√(m²a² + b²));

Answer 2

The derivative by implicit differentiation is dy/dx=-b²x/(a²y).

There will be 2 places where this derivative equals m.