Equation of parabola3

In this page 'Equation of parabola 3' we are going to find the equation of the parabola for the given information.

Question 3:

Find the equation of the parabola whose vertex is the origin(0,0) and the equation of the directrix is x = 2.

Solution:

Method I Here the  given vertex is the origin (0,0) and the directrix is x=2.

We know that vertex is halfway between the focus and the directrix. Since the vertex is the origin and directrix is x=2 which is a vertical line, the parabola is opening on the left side.

The focus S is (-2,0)

Let P(x,y) be a point on the parabola.

If we draw a line PM perpendicular to the directrix, we know that

SP/PM = 1. (as it is a parabola, SP/PM=e=1)

SP²    =  PM²

(x+2)²+(y-0)² = [(x-2)/√(1²+0²)}]²

x²+4-4x+y²    =  (x-2)²/1

x²-4x+4+y²    =   x²-4x+4

Simplifying we get

y²     = -8x

Method II Equation of parabola3 Equation of parabola3 Equation of parabola3 Equation of parabola3

We know that vertex is half way between the focus and directrix. We also know that the parabola curves away from the directrix. The vertex is the origin(0,0). Since the parabola opens in the left hand side 'a' must be negative and the value of a is the distance between the focus and vertex and the distance between the vertex and directrix. So here the value of a is -2

So the equation of the parabola in the vertex form is

(y-k)² = 4a(x-h)

(y-0)² = 4(-2)(x-0)

y²     = -8x

Students can follow either method to derive the equation of the parabola. Parents and teachers can guide the students to understand both methods of 'Equation of parabola 3' and guide them to do the practice problems using one of the above methods.

 Practice Questions Solution (1) Find the equation of the parabola whose focus is (3,0) and the equation of the directrix is x=-3. Solution (2) Find the equation of the parabola whose focus is (4,1) and the equation of the directrix is x = 0. (4) Find the equation of the parabola whose vertex is (1,2) and the equation of the directrix is y = 0. (5) Find the equation of the parabola whose vertex is (-2,-1) and the focus is (-4,-1). (6) Find the equation of the parabola whose vertex is the origin (0,0) and the focus is (0,4).

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