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.
The directrix will cross through the positive part of the y-axis.
The equation of the parabola will be in the form y2 = 4px where the value of p is negative.
The equation of the parabola will be in the form x2 = 4py where the value of p is positive.
The equation of the parabola could be y2 = 4x.
The equation of the parabola could be x2 = y.
Solution:
Given, focus of parabola on the negative part of the y-axis and vertex at origin.
This implies that the parabola will be a vertical parabola and opens downward.
The general equation of vertical parabola is
\((x-h)^{2}=4p(y-k)\)
Where, (h, k) is the vertex point
Here, (h,k) = (0,0)
The equation becomes,
\((x-0)^{2}=4p(y-0)\)
\(x^{2}=4py\)
1) As the parabola opens downward, so the vertex is the highest point and the directrix line will be above the vertex.
As the vertex is at (0,0) so the directrix will cross through the positive part of the y-axis.
Therefore, option (1) is true
2) The general equation of the parabola is \(x^{2}=4py\).
So, option (2) is not true.
3) As the axis of symmetry is the negative y-axis, the value of ‘p’ in the equation \(x^{2} = 4py\) will be negative.
Thus, option (3) is not true.
4) \(y^{2}=4x\) is like the form \(y^{2}=4px\).
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 = 1/4 which is a positive value.
Here, the value of p must be negative.
Hence, option(5) is not true.
Therefore, the directrix will cross through the positive part of the y-axis is true.
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?
Summary:
A parabola, with its vertex at (0,0), has a focus on the negative part of the y-axis.The directrix will cross through the positive part of the y-axis.