Question

Find the coordinates of the focus, the equation of the directrix, the length of the straight side, and the graph of the Parabola whose equation is: -3x2 – 12y = 0 -2y2 + 9x = 0

Answer 1

Answer:

The given equation of the parabola is y2 - 4x - 4y = 0

⇒ y2 - 4y = 4x

⇒ y2 - 4y + 4 = 4x + 4, (Adding 4 on both sides)

⇒ (y - 2)2 = 4(x + 1) ……………………………….. (i)

Shifting the origin to the point (-1, 2) without rotating the axes and denoting the new coordinates with respect to these axes by X and Y, we have

x = X + (-1), y = Y + 2 ……………………………….. (ii)

Using these relations equation (i), reduces to

Y2 = 4X……………………………….. (iii)

This is of the form Y2 = 4aX. On comparing, we get 4a = 4 ⇒ a = 1.

The coordinates of the vertex with respect to new axes are (X = 0, Y = 0)

So, coordinates of the vertex with respect to old axes are (-1, 2), [Putting X= 0, Y = 0 in (ii)].

The coordinates of the focus with respect to new axes are (X = 1, Y = 0)

So, coordinates of the focus with respect to old axes are (0, 2), [Putting X= 1, Y = 0 in (ii)].

Equation of the directrix of the parabola with respect to new axes in X = -1

So, equation of the directrix of the parabola with respect to old asex is x = -2, [Putting X = -1, in (ii)].

Equation of the axis of the parabola with respect to new axes is Y = 0.

So, equation of axis with respect to old axes is y = 2, [Putting Y = 0, in (ii)].

The length of the latusrectum is 4 units.