Question

If the coordinates of the vertex and focus of a parabola are (-1,1) and (2,3) respectively then write the equation of its directrix.

Answer 1

Given that,

We need to find the equation of the directrix of a parabola whose focus is (2, 3) and having a vertex at (-1,1).

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 the axis. We know that the slope of the straight line passing through the points (x₁, y₁) and (x₂, y₂) is \(\frac{y_2-y_1}{x_2-x_1}\).

We know that,

The product of slopes of the perpendicular lines is - 1. 

Let m₂ be the slope of the directrix. 

⇒ m₁.m₂ = - 1

⇒ \(\frac{2}{3}\) x m₂ = -1

⇒ m₂ = \(\frac{-3}{2}\)

Let us assume the intersection point on directrix is (x₁, y₁).

⇒ x + 2 = - 2 and y + 3 = 2 

⇒ x = - 4 and y = - 1. 

The point on directrix is (-4,-1). 

We know that,

Equation of the straight line passing through point (x₁, y₁) and slope m is y - y₁ = m(x - x₁).

⇒ y - (-1) = \(\frac{-3}{2}\)(x - (-4))

⇒ 2(y + 1) = - 3(x + 4) 

⇒ 2y + 2 = - 3x - 12 

⇒ 3x + 2y + 14 = 0 

∴The equation of the directrix is 3x + 2y + 14 = 0.