Normal at a point of an ellipse:

Equation of normal is 

Equation of normal at (a cosθ, b sinθ) is 

Example: If the normals to ellipse at points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are concurrent, show that 

Solution:  The equation of normal to the given ellipse at (x₁, y₁) is

            a²xy₁ - b²yx₁ - (a² - b²)x₁y₁ = 0                 . . .(1)

           Likewise the normals at (x₂, y₂) and (x₃, y₃) are

            a²xy₂ - b²yx₂ - (a² - b²)x₂y₂ = 0                 ...(2)

            a²xy₃ - b²yx₃ - (a² - b²)x₃y₃ = 0                 ...(3)

Eliminating a²x, b²y and (a² - b²) from (1), (2) and (3), we find that 3 lines are concurrent if

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