In this page 'Equation of parabola 4' we are going to find the equation of the parabola for the given information.
Question 4:
Find the equation of the parabola whose vertex is (1,2) and the equation of the directrix is y = 0.
Solution:
Method I
Here the given vertex (1,2) and the directrix is y=0.
We know that vertex is halfway between the focus and the directrix. Since the vertex is (1,2) and directrix is y=0 which is x axis, the parabola is opening up.
The focus S is (1,4)
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-1)²+(y-4)² = [(y-0)/√(1²+0²)}]²
x²+1-2x+y²+16-8y = y²/1
x²-2x+y²-8y+17 = y²
Simplifying we get
y² -2x-8y+17=0
Method II
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 (1,2). Since the parabola opens up 'a' must be positive 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
(x-h)² = 4a(y-k)
(x-1)² = 4(2)(y-2)
Simplifying,
x²-2x+1= 8y-16
x²-2x-8y+17 = 0
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 4' 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. | |
(3) Find the equation of the parabola whose vertex is the origin (0,0) and the equation of the directrix is x = 2. | |
(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). |
Equation of parabola4 Equation of parabola4 |