equation of parabola

Question

write the equation of the parabola in standard form using the given information:
focus : (-1,2) & directrix: x=3

Answer 1

Given information focus: (-1, 2) and directrix: x = 3.

The equation of the parabola in standard form is (x - h)² = 4p(y - k)

where (h,k) is the vertex, p is the distance from the vertex to the focus and also the distance from the vertex to the focus.

the focus F(-1, 2). I'll put an "o" there. And I'll draw the directrix x = 3 which is a horizontal line 3 units above the y-axis. I'll draw it green The vertex is halfway between the focus and the directrix. So the vertex is at the point V(4, 2).

That means that the vertex (h,k) = (4, 2)

So the distance from V(4, 2) to F(-1, 2) is 5 units, so p = 5.

In the equation (x - 4)² = 4p(y - 2)

We can put in 5 for p and have (x - 4)² = 4(5)(y - 2)

(x - 4)² = 20(y - 2)

The equation of the parabola is (x - 4)² = 20(y - 2).

Comments

The above solution is wrong.

See the correct solution below .

Answer 2

Focus of parabola (-1,2) and directrix x  = 3

If a parabola has a horizontal axis, the standard form of the parabola is (y - k )² = 4p (x - h )

Where p not equals to 0.Vertex (h,k ) ,focus (h +p , k ) and directrix x  = h - p

(h +p ,k ) = (- 1,2)

h + p  = -1, k  = 2

h + p  = -1 ---> (1)

Directrix x  = h - p = 3

h - p = 3 ---> (2)

h  = 3 + p

Substitute the h  value in equation (1).

3 + p + p  = - 1

2p  = - 4

p  = - 2

substitute p  value in equation (1).

h - 2 = -1

h  = 1

So the vertex is (1 , 2)

p is negitive the parabola opens to the left.

Substitute the values of h ,k and p in standard form of parabola.

Equation of parabola is (y - 2)² = -8(x - 1).