Find the point(s) on the ellipse: x²/a² + (y - k)/b² = 1 where the tangent line that passes through the point: (m, n)?

first x²/a² + (y − k)/b² = 1 is a parabola not an ellipse so let's assume you meant x²/a² + (y − k)²/b² = 1

x²/a² + (y − k)²/b² = 1

∴ d/dx(x²/a² + (y − k)²/b²) = d/dx(1)

∴ d/dx(x²/a²) + d/dx((y − k)²/b²) = d/dx(1)

∴ (1/a²)d/dx(x²) + (1/b²)d/dx((y − k)²) = d/dx(1)

∴ (1/a²)d/dx(x²) + (2/b²)(y − k) * d/dx(y − k) = d/dx(1)

∴ (1/a²)d/dx(x²) + (2/b²)(y − k)(d/dx(y) − d/dx(k)) = d/dx(1)

∴ 2x/a² + (2/b²)(y − k)(dy/dx − 0) = 0

∴ (2/b²)(y − k)(dy/dx) = - 2x/a²

∴ dy/dx = -b²x/(a²(y − k))

∴ dy/dx = b²x/(a²(k − y))

given a tangent line at (x₁,y₁) is equal (y − y₁) = (dy/dx)(x − x₁)

so at (n,m) it is (y − n) = (b²m/(a²(k − n)))(x − m)

∴ y = b²m(x − m)/(a²(k − n))) + n