Question

A parabola, with its vertex at (0,0), has a focus on the negative part of the y-axis.

Which statements about the parabola are true? Check all that apply.

1: The directrix will cross through the positive part of the y-axis.
2: The equation of the parabola will be in the form y2 = 4px where the value of p is negative.
3: The equation of the parabola will be in the form x2 = 4py where the value of p is positive.
4: The equation of the parabola could be y2 = 4x.
5: The equation of the parabola could be x2 = y.

Answer 1

The only true statement is :

1: The directrix will cross through the positive part of the y-axis

Explanation

Vertex of the parabola is at (0,0) and the focus is on the negative part of the y-axis. That means, the parabola will be a vertical parabola and opens downward.

The general equation of a vertical parabola is: $(x-h)^2 =4p(y-k)$ , where (h, k) is the vertex point.

Here, vertex is (0, 0), so the equation will be:

$(x-0)^2 = 4p(y-0)\\ \\ x^2 = 4py$

1. As the parabola opens downward, so the vertex is the highest point and the directrix line will be above the vertex. As vertex is at (0, 0), so the directrix will cross through the positive part of the y-axis. That means, option (1) is true.

2. The general equation of the parabola is like: $x^2 = 4py$ . So option (2) is not true.

3. As the axis of symmetry is the negative y-axis, so the value of 'p' in the equation $x^2 = 4py$ will be negative. That means, option (3) is not true.

4. $y^2 = 4x$ is like form of $y^2 = 4px$. But here the general form of the parabola is: $x^2 = 4py$. So, option (4) is not true.

5. If we compare the equation $x^2 = y$ with the general form $x^2 = 4py$, then we will get:

$4p= 1 \\ p= \frac{1}{4}$ which is a positive value

But here the value of 'p' must be negative. So, option (5) is also not true.

Answer 2

Answer:

options 1, and 5 are correct

Step-by-step explanation:

I dont have a explanation but i just did the test and it gave me the answers...