Question

Find the vertex, axis, focus, directrix, latusrectum of the parabola, also draw their rough sketches 4y² + 12x – 20y + 67 = 0

Answer 1

The given equation is 

4y² + 12x – 20y + 67 = 0

Using these relations, equation (i) reduces to Y2 = – 3X ....(iii) 

This is of the form Y2 = – 4aX. 

On comparing, we get 4a = 3 

⇒ a = 3/4. 

Vertex - The coordinates of the vertex are (X = 0, Y = 0) So, the coordinates of the vertex are 

(- 7/2, 5/2)

[Putting X = 0, Y = 0 in (ii)] 

Axis: The equation of the axis of the parabola is Y = 0. So, the equation of the axis is 

y = 5/2    [Putting Y = 0 in (ii)] 

Focus- The coordinates of the focus are (X = –a, Y = 0) 

i.e. (X = – 3/4, Y = 0). 

So, the coordinates of the focus are 

(–17/4, 5/2)   [Putting X = 3/4 in (ii)]

Directrix - The equation of the directrix is X = a i.e. X = 3/4.

So, the equation of the directrix is

 x = - 11/4    [Putting X = 3/4 in (ii)] 

Latusrectum - The length of the latusrectum of the given parabola is 4a = 3.