Given that we need to find the equation of the ellipse passing through the points (1,4) and (- 6,1).

Let us assume the equation of the ellipse as (a²>b²). ..... .... (1)

Substituting the point (1,4) in (1) we get

⇒

⇒

⇒

⇒ b² + 16a² = a² b² ..... - - (2)

Substituting the point (- 6,1) in (1) we get

⇒

⇒

⇒

⇒ a² + 36b² = a²b² ..... - - (3)

(3)×16 - (2)

⇒ (16a² + 576b²) - (b² + 16a²) = (16a²b² - a²b²)

⇒ 575b² = 15a²b²

⇒ 15a² = 575

⇒

From (2)

⇒

⇒

⇒

The equation of the ellipse is

⇒

⇒

⇒ 3x² + 7y² = 115

∴ The equation of the ellipse is 3x² + 7y² = 115.