Write the c

Given that we need to find the equation of the parabola whose focus is S(- 2, 1) and directrix(M) is x + y - 3 = 0.

We know that the directrix is perpendicular to the axis and vertex is the midpoint of focus and the intersection point of axis and directrix.

Let us find the slope of directrix. We know that the slope of the straight line ax + by + c = 0 is .

⇒

⇒ m₁ = - 1

We know that the product of slopes of the perpendicular lines is - 1.

Let m₂ be the slope of the directrix.

⇒ m₁.m₂ = - 1

⇒ - 1×m₂ = - 1

⇒ m₂ = 1

We know that the equation of the straight line passing through (x₁, y₁) and having slope m is y - y₁ = m(x - x₁).

⇒ y - 1 = 1(x - (- 2))

⇒ y - 1 = x + 2

⇒ x - y + 3 = 0

On solving the lines x - y + 3 = 0 and x + y - 3 = 0 we get the intersection point to be (0, 3).

Let us assume the vertex be (x₁, y₁).

⇒

⇒ (x₁, y₁) = (- 1, 2)

∴The coordinates of the vertex is (- 1, 2).