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. | |
(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). |
Quote on Mathematics
“Mathematics, without this we can do nothing in our life. Each and everything around us is math.
Math is not only solving problems and finding solutions and it is also doing many things in our day to day life. They are:
It subtracts sadness and adds happiness in our life.
It divides sorrow and multiplies forgiveness and love.
Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?
Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”